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About Government News...
Transport Topics government and regulatory coverage keeps managers of a highly-regulated industry aware of the policy decisions that can shape their businesses. Covering both the legislative and regulatory aspects of policy-making, at both the state and national levels, the news in this category includes looks at infrastructure, hours of service, emissions rules, funding measures, leadership appointments, and more. Readers can follow what's happening in Congress, at the Department of Transportation and the Federal Motor Carrier Administration, and in state and local governments.
Business, Safety, Government, Equipment, Autonomous
ArvinMeritor to Develop Diesel-Electric Drivetrain for Wal-Mart
Trucking component maker ArvinMeritor and Wal-Mart Transportation have agreed to develop a dual-mode, diesel-electric drivetrain for a Class 8 tractor, the two said.
Podcast Debut: 'RoadSigns' Tackles Autonomous
Welcome to the inaugural episode of Transport Topics' new podcast: RoadSigns. Our debut episode focuses on how the trucking industry's steady and ongoing development of autonomous technology may someday affect truck drivers. Click to listen!
Business, Fuel, Government
Schwarzenegger Orders Calif. to Further Cut Fuel Emissions
California Gov. Arnold Schwarzenegger (R) plans to ask state regulators to require the state's petroleum refiners and gasoline sellers to cut greenhouse-gas emissions by 10%, the New York Times reported Wednesday.
Business, Government, Autonomous
Trucking-Related Revenue Rose to $292 Billion in '05
Trucking, couriers and messengers, and warehousing and storage revenue reached $292 billion in 2005, up from $266 billion the year before, the U.S. Census Bureau reported.
Paccar to Build $400 Mln. Powertrain Plant in Southeastern U.S.
Paccar Inc., the parent company of Kenworth Truck Co. and Peterbilt Motors Co., said late Tuesday it plans to build a new $400 million powertrain manufacturing plant in the southeastern United States.
Business, Government
Wholesale Inventories, Sales Gain for November
Wholesale inventories jumped 1.3% in November as sales rebounded from two consecutive declines, rising 1%, the Commerce Department said Wednesday.
U.S. Trade Deficit Narrowed in November, Commerce Reports
The U.S. trade deficit narrowed in November to its smallest level since July 2005 and, as exports reached a record and oil imports declined, the Commerce Department reported Wednesday.
Business, Safety, Government, Autonomous
Security & Safety Briefs — Jan. 4 - Jan. 10
Highway Watch Report Leads to Arrest • McMurray Named FMCSA's Chief Safety Officer • Mass. Trucking Company Files Complaint Against FMCSA • Survey Cites Drivers' HOS Compliance Concerns • And more...
Business, Government, Safety, Autonomous
FMCSA to Propose Electronic Onboard Recorder Rule Thursday
The Federal Motor Carrier Safety Administration said late Tuesday that it will announce a proposed rule regarding electronic on-board recorders for commercial trucks and buses on Thursday morning.
Business, Fuel, Government, Autonomous
Crude Oil Closes at 19-Month Low, at $54.02 a Barrel
Crude oil futures hit a 19-month low price, closing at $54.02 a barrel on the New York Mercantile Exchange Wednesday following a Department of Energy report showing strong heating-oil inventories, Bloomberg reported.
Business, Government, Safety
ATRI Issues Final Request for HOS Sleeper-Berth Data
The American Transportation Research Institute issued a final call Tuesday for motor carrier safety data to measure the effects of the federal hours-of-service rule's split sleeper-berth provisions.

redpajama_set_name: RedPajamaCommonCrawl


Dr. K. Rajeswara Rao was inducted to the Board as an Additional director w.e.f. 19.02.2018.
Shri Anil Kumar Nayak was inducted to the Board as an Additional director w.e.f. 27.03.2018.
Born on 16.05.1962, Shri Anil Kumar Nayak has done his M.A. in Politics from School of International Studies, JNU, New Delhi and M.Phil in Ethnicity and Political development in the Ivory Coast from Delhi University. He joined Indian Ordinance Factory Services (IOFS-1986 Batch) on 15th Feb' 1987. He was Deputy Director General, Ordinance Factory Board, Kolkata looking after Industrial Relations of the entire organization comprising of 41 Ordinance factories before joining as Chief Labour Commissioner in Ministry of Labour in August, 2015. He got wide experience of working in other Government departments viz. Under Secretary to 5th Pay Commission, Deputy Secretary in Ministry of Defence, Regional Provident Fund Commissioner Gr.-I of Bihar and Odisha, Director in Ministry of Mines. He has been appointed as Joint Secretary, Ministry of Mines in March, 2018. He brings with him varied experience in technical as well as administrative fields.
Shri V. Balasubramanyam joined the Company as Director (Production) w.e.f. 01.01.2015.
Born on 01.12.1960, Shri V Balasubramanyam completed his B.Tech in chemical engineering and joined NALCO as a Graduate Engineer Trainee (GET) in 1984. During his long service association of three decades with NALCO, Shri Balasubramanyam has contributed significantly from Technology adoption to absorption in the field of aluminium technology. With his vast professional experience, which nested from project execution to plant operation at both the production complexes of NALCO, Shri Balasubramanyam held very critical and crucial positions in the organization before taking over as Director (Production).
Shri Balasubramanyam is a life member of Indian Institute of Metals (IIM), Management Committee member of Federation Indian Mineral Industries (FIMI) and also member of Energy Panel in Odisha chapter of the Confederation of Indian Industries (CII).
Shri Basant Kumar Thakur is the Director (HR) of the Company w.e.f. 04.07.2016.
Born on 19.12.1959, Shri Basant Kumar Thakur has a Diploma in Social Works along with a degree in History from Punjab University. He started his career in SAIL in 1981 and since then he served in various units, including Durgapur Steel Plant, Bokaro Steel Plant, Salem Steel Plant, R&D Centre in Ranchi and Corporate Office in New Delhi, prior to joining NALCO as Director (HR). Shri Thakur is a hands on Human Resources professional with comprehensive experience including recruitment, retention, conflict resolution, change management, labour relation and benefit administration. He has four years of experience in Corporate Communications. In SAIL, he had collaborated with senior management to conduct organizational wide strategic planning in order to support and further the organizational goals. He played a vital role rebuilding Human Resources Department by updating Human Resources System, policies and procedure development, coaching, counseling, planning, direction and management of all HR activities in his three decades long career at SAIL. Shri Thakur is a life member of NIPM.
Shri Sanjib Kumar Roy is the Director (P&T) of the Company w.e.f. 03.02.2017.
Shri Roy completed B Sc (Hons in Chemistry) from Ramakrishna Mission Vidyamandir, Belurmath and then B Tech & M. Tech in Chemical Engineering discipline from Calcutta University. He began his career in NALCO as a Graduate Engineer Trainee in 1984. He was posted in the company's Alumina Refinery Complex in Damanjodi since inception of the project, where he held different key positions including two stages of expansion before becoming the General Manager (Refinery). Thereafter in May'2012, he was posted as General Manager (Smelter) at the company's Smelter Plant in Angul before his elevation as Executive Director (S&P). Shri Roy moved to the headquarters in Bhubaneswar as Executive Director (Production) in April 2015.
Shri Roy brings to this post his vast experience in the company's plant & operations as well as managing projects from conceptualization to commissioning.
Shri Pradip Kumar Mishra Joined the Company as Director (Commercial) w.e.f. 23.04.2018.
Born on 12.02.1961, Shri Pradip Kumar Mishra is a Post Graduate in English literature from Utkal University. He started his professional career as a Management Trainee in SAIL in the year 1983. He brings along rich and varied experiences in the field of marketing management. During his 35 years tenure in SAIL, he served in various positions, including being the Regional Manager in three regions of SAIL. He headed the domestic marketing of flat products of SAIL for over 3 years as Executive Director. Shri Mishra is the recipient of the prestigious Jawahar Award for outstanding contribution in marketing area in SAIL.
Born on 12.10.1964, Shri Patra is a graduate in Commerce from Utkal University and a Member of Institute of Chartered Accountants of India. He has also done MBA (HRD) from Vidya Sagar University. He has more than three decades of experience in various Public Sector Undertaking like Odisha Mining Corporation Ltd., Indian Rare Earths Ltd. and Mangalore Refinery & Petrochemicals Ltd. (A subsidiary of ONGC Ltd.). He has contributed as an academician apart from his professional employment in PSUs. Before joining NALCO as Director (Finance), he served as Director (Finance) of THDC India Limited from 02.08.2013 to 31.08.2018.
Prof. Damodar Acharya was inducted to the Board as an Independent Director on 21.11.2015.
Born on 2nd April, 1949, Prof. Damodar Acharya holds degree in Mechanical Engineering from NIT, Rourkela, Masters and PhD degrees from IIT, Kharagpur. He joined the Industrial Engineering faculty in the same institute in 1976. He has left his indelible mark in all the responsibility that he has taken in the institute, be it as head of the Department, Chairman JEE, DEAN (Sponsored Research and Industrial Consultancy), Executive Director STEP, Chairman of Vinod Gupta School of Management or as the Director of this first and the largest IIT in the Country.
As Founding Vice Chancellor of Biju Patnaik University of Technology, he laid a foundation of a robust Technical University Education system that is being emulated by others. He was the Chairman of All India Council of Technical Education.
He played important role in the establishment of IIT Bhubaneswar and was its first Mentor Director. He remained non official Director in the Central Board of RBI for four years and as an Independent Director in RCF for three years.
Prof. Acharya is a Member of Chhatisgarh State Planning Commission. He is also in the Boards of IDCO and State Pollution Control and Chairman of the Advisory Board of SOA University.
Shri Dipankar Mahanta was inducted to the Board as a Part-time Non-official (Independent) Director w.e.f. 21.11.2015.
Born on 12th December, 1965, Shri Dipankar Mahanta did his MBA from Guwahati University and started with an entrepreneurial venture named M/s Consort Marketing, with an objective of marketing small scale industries products. Subsequently, he joined and served the Guwahati Stock Exchange in various capacities concerning the Indian Capital Market. He was the promoter director of Economic and Industrial Development Collaborative (India) Pvt. Ltd. a company with an object for imparting quality consultancy services and its implementation in the North East Region (NER). He was a consultant in designing and implementing a Handmade Paper unit for L.B.Agro Private Ltd near Guwahati. He was involved as co-consultant in projects like Diagnostic Study of Industrial Training Institutes (ITI's) of the NER for the NEC, Feasibility Study of a Market Complex Exclusively Selling Handloom and Handicraft Products for NEHHDC, Study of the factors affecting the Process of Language Learning for DPEP, Study on 'Bamboo and Bamboo Products in Assam' for National Bamboo Mission, Study on 'Socio-economic Study of the Brick Field in Assam' for Pollution Control Board, Assam. He served Vivekananda Kendra, a voluntary organization in various capacities and later on he was the Associate Director, Research Council of the Vivekananda Kendra Institute of Culture (VKIC), a specialized project on Cultural Documentation and Research of India's North East. He also had the opportunities to serve CAPART (NEZ) under the Ministry of Rural Development, Government of India.
He is presently working in the field of Social Entrepreneurship with various social organizations, prominent amongst them is Vivekananda Kendra and Srimanta Foundation for Culture and Society. He is also a resource person and trainer for sessions on 'Knowledge of Self and its Management '.
Shri Maheswar Sahu was inducted to the Board as a Part-time Non-official (Independent) Director w.e.f. 21.11.2015.
Shri Maheswar Sahu has done B.Sc. (Engg.) in Electrical in 1977 from NIT, Rourkela and M.Sc. from University of Birmingham in 1994. He joined Indian Administrative Service (IAS) in 1980. He has served the Government of India and Govt of Gujarat in various capacities for more than three decades before retiring as Additional chief secretary, Govt. of Gujarat in 2014. His career span includes more than 20 years of service in industry and more than 10 year of active involvement in PSU management. He had worked more than 3 years in United Nations Industrial Development Organization. He was instrumental in organization of four Vibrant Gujarat events.
He served as director in many CPSEs. He was also Chairman/ Director in many State PSUs. He is now associated with a number of companies as Independent Director/Chairman. His area of specialization includes strategic management, public administration, corporate governance etc.
Shri S. Sankararaman was inducted to the Board as a Part-time Non-Official (Independent) Director from 21.11.2015.
Born on 19th May, 1962, Shri S. Sankararaman is a fellow member of the Institute of Chartered Accountant of India (ICAI). Shri S. Sankararaman is affected by muscular dystrophy and a wheelchair user from the age of 12. He is presently the Honorary Secretary of Amar Seva Sangam, an Institution for betterment of the handicapped and under privileged. He started his career in 1985 rendering professional services to various corporates before joining the Amar Seva Sangam in 1992. His mission is to empower the disabled citizen by establishing a "Valley for the Disabled" as a Rehabilitation and Development Centre for the region and developing models for self-help initiatives by integrating the disabled individuals with the society for improved living conditions in the villages.
He is a champion for the rights of the disabled persons and believes that disability is not a constraint but only a condition which can be managed with the right combination of rehabilitation and enablement. He has been instrumental in developing Amar Seva Sangam from scratch to its current position of leadership in the field of disability management with state-of-the-art infrastructure and deliver high quality rehabilitation programs across all age groups and across all disabilities. He is a pioneer in developing innovative models for rehabilitation of persons with disabilities with the involvement of community and has spearheaded a Village Based Rehabilitation Initiative which is acknowledged to be one of the best models in the country. He believes in professionalizing the NGO sector and has computerized all the activities of Amar Seva Sangam with systems and processes. He organized national level seminar on law on handicapped resulting in the enactment of Persons with Disabilities Act by the Indian Parliament in 1996 and many other. He participates in various marathons particularly the Standard Chartered Mumbai Marathon from its inception and has paved way for the inclusion of disabled persons in such sporting events. He has presented papers on the rights of disabled in various forums across the country. He is a trustee and board member of several nonprofits including the Gandhigram Trust (Treasurer), Gandhigram, Dindigul, the Collective action for Basic Rights Foundation (Hon. Vice President), Bangalore, Tamil Nadu Udavikkaram Association for Differently Abled, (Founder Member), Chennai, National Institute for Empowerment of Persons with Multiple Disabilities (Member) and from 5th May 2017 he has been appointed as a Member, Tamil Nadu State Commissionerate for the Differently Able, Chennai, to represent persons with loco-motor disabilities. He was conferred with the Ashoka Fellowship for innovation and received several awards at the State, National and International levels in recognition of his services, which include a State Award from the Chief Minister of Tamil Nadu and National Awards from the President of India twice.
Shri Pravat Keshari Nayak was inducted to the Board as a Part-time Non-Official (Independent) Director from 21.11.2015.
Shri Pravat Keshari Nayak, Chartered Accountant by profession is senior partner of leading CA Firm M/s. P. K. Nayak & Co., rendering comprehensive professional services to Government and private sectors. With a career spanning over three decades, his vast repertoire of financial expertise and experience spans across the public and private sectors and several corporates, banks and other financial institutions. He has consulted for Adam Smith International, UK on public sector reforms for restructuring of PSUs in Odisha. He also holds degrees in Laws and diploma in Information System Audit (ICAI).
Besides undertaking audits for government and as Independent auditor in large social sector projects, he has proactively offered guidance to young entrepreneurs and start-ups. He has also been a part of many national seminars and conferences related to Finance, taxation and audits.
Ms. Kiran Ghai Sinha was inducted to the Board as a Part-time non-official (Independent) director w.e.f. 03.02.2017.
She was senate Member, Patna University for three terms in succession. She is also Chairperson, Scouts and Guides Fellowship, Bihar. She is member (Non Govt.), Hindi Salahkar Samiti, Ministry of Civil Aviation, Govt of India.
Ms. Sinha did her M.A. in Hindi from Patna University. She retired as an Associate Professor, Dept. of Hindi, Patna Women's college, Patna University. Ms. Sinha was also member, Bihar Legislative Council (BLC) for consecutive two terms from 2004-10 and again 2010-16. During her tenure as MLC, she was privileged to chair 3 Committees of the house viz. Urban development Committee, Rajbhasha Committee & Child protection Women empowerment Committee. She was member, Local Board (Eastern Zone), Reserve Bank of India (2001-2004) and was also member, Managing Board, Rajendra Agriculture University, Pusa, Bihar for two terms. She was also a jury member for non feature films category & feature film category in the 48th and 50th National Film awards. She was a member of a state delegation to the World Hindi Conference held at United Nations, USA (2007).
Shri N N Sharma was inducted to the Board as a Part-time nonofficial (Independent) director w.e.f. 06.09.2017.
Born on 12th August, 1950. Shri N N Sharma has done B.Sc in Mechanical Engineering. At present, he is Chairperson, Centre for Corporate Social Responsibility and Sustainability at Birla Institute of Management Technology (BIMTECH), Greater Noida. He is also member of Social Audit Committee of ACC Ltd. Shri Sharma conducted capacity building programme of CSR professionals of corporates like NTPC, NHPC, SAIL and NPCIL. He has more than three decades of experience of working in social development sector. He was managing director of Rajasthan Consultancy Organisation (RAJCON) promoted by IFCI and other development organizations and also first general manager of UP Minorities Financial and Development Corporation.
He has also worked with UNDP and UNIDO and was associated with projects supported by the World Bank and DFID, UK.
Smt. Achla Sinha was inducted to the Board as a Part-time non-official (Independent) Director w.e.f. 08.09.2017.
Smt. Achla Sinha has done M.A. in English Literature from Patna University, M.Phil in Public Administration from Punjab University, Advance Professional Programme in Public Administration (APPA) from Indian Institute of Public Administration (IIPA), New Delhi.
She is an Indian Railways Traffic Service officer (IRTS) of the 1982 Batch. She started her career from Eastern Railway, Danapur Division as an Assistant Commercial Manager in 1984 and had since worked in various capacities for over three decades in the Ministry of Railways as Director (Tourism), Executive Director (Statistics & Economics) and in the Delhi Division of Northern Railway as Chief Traffic Manager. She also served as Additional Secretary, Central Information Commission prior to her superannuation in October, 2016. She has won the General Manager Eastern Railway Award during the Annual Railway Week for outstanding performance and the Best Woman Participant in the 30th Advance Professional Programme in Public Administration (APPA) from IIPA, New Delhi. She has written a number of concept papers to bring about improvements in Indian Railway's functioning as also in Urban Development Ministry, Government of India. Many of these concepts have since been discussed and implemented.


The Dodos Day Fest Venue Change!
We have recently been informed that alcohol selling isn't allowed at the Siam Society, therefore SUPERSWEET Live! presents the Dodos Day Fest will move to Centerpoint Studio, Sukhumvit 105. We are very sorry for this inconvenience.
Centerpoint Studio is an awesome space, it might seem a little big for a show of 400 capacity. But this time, we are planning to place the performance and activity areas indoor so we can all enjoy the air-con breeze. And we promise to keep the show as intimate and sweet as we first promised!
As for the mentioned activities, we will keep it short with this list: picnicking, crystal psychic reading, the hula-hoop contest, tote-painting workshop, shops from SS and bands, DJ booth by Coconut Project (who are performing, we were told!), and very very nice food stall by Minibar Royale.
Most importantly, don't leave after the show, the Dodos will set up a signing tent where you can pre-purchase their imported CDs, vinyls and our very cool totes and tees. Get signed!


Waseca Bluejays - Case for iPhone X/XS/Max/XR – Twig Case Co.
Twig Case Co. is located in Waseca, Minnesota, and the Bluejays are our hometown team - may the odds be ever in our favor!
Congrats to the Waseca Basketball team for this amazing win!
These cases are made from paper? Yep.


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\section{Introduction}

A ring $\Lambda$ is said to be {\it of finite representation type} or {\it representation-finite} if it is left artinian and there are only finitely many finitely generated indecomposable  left $\Lambda$-modules up to isomorphism. The class of representation-finite rings is one of the most important class of rings. Auslander in his famous theorem, which is called Auslander correspondence, provided a bijection between the set of Morita equivalence classes of representation-finite rings and that of rings with nice homological properties. Let $\Gamma$ be an artinian ring. The {\it global dimension} of $\Gamma$, which is denoted by ${\rm gl.dim}(\Gamma)$, is at most $n$ if the projective dimension of each left $\Gamma$-module is less than or equal $n$. The dominant dimension of $\Gamma$, which is denoted by ${\rm dom.dim}(\Gamma)$, is at least $n$ if the first $n$ terms in the minimal injective resolution of $_{\Gamma}\Gamma$ are projectives. An artinian ring $\Gamma$ is called an {\it Auslander ring} if  ${\rm gl.dim}(\Gamma) \leq 2$ and ${\rm dom.dim}(\Gamma) \geq 2$. According to the Auslander correspondence  \cite[Corollary 4.7]{ausla2}, there exists a bijection between the set of Morita equivalence classes of representation-finite rings and that of Auslander rings $\Gamma$. It is given by $\Lambda\rightarrow\Gamma:=\rm{End}_{\Lambda}(M)$, where $M=\bigoplus_{i=1}^nM_i$ and $\lbrace M_1, \cdots, M_n\rbrace$ is a complete set of representative of the isomorphic classes of finitely generated indecomposable left $\Lambda$-modules. $M$ is called the Auslander generator of $\Lambda$-mod. Auslander correspondence gives a very nice connection between concept of representation-finiteness which is a representation theoretic property and the concept of Auslander ring which is a homological property. On the other hand, the category Mod-$\Gamma$ is equivalent to the functor category Mod$((\Lambda$-mod$)^{\rm op})$, where Mod$((\Lambda$-mod$)^{\rm op})$ is the category of all additive covariant functors from $\Lambda$-mod to the category of abelian groups. This equivalence propose to use the functor categories for studying the representation-finite rings which leads
to functorial approach in representation theory (see \cite{Au, ausla2}).
A functor $F\in$ Mod$((\Lambda$-mod$)^{\rm op})$ is called noetherian (resp., artinian) if it satisfies the
ascending (resp., descending) chain condition on subfunctors. $F$ is called finite if it is both noetherian and artinian and $F$ is called locally finite if every finitely generated
subfunctor of F is finite. The category  Mod$((\Lambda$-mod$)^{\rm op})$ is said to be locally finite if every $F\in$ Mod$((\Lambda$-mod$)^{\rm op})$ is locally finite. Auslander in Theorem 3.1 of \cite{ausla2} proved that a ring $\Lambda$ is of finite representation type if and only if Mod$((\Lambda$-mod$)^{\rm op})$ is locally finite if and only if every representable functor of Mod$((\Lambda$-mod$)^{\rm op})$ is both artinian and noetherian (see also \cite{aus}).

It is known that every finitely generated ${\Bbb Z}$-module decompose to the finite direct sum of cyclic modules. Artinian rings with this property are representation-finite rings. These rings are precisely rings over which every module is a direct sum of cyclic modules. A ring $\Lambda$ is called {\it left K\"othe} if every left $\Lambda$-module is a direct sum of cyclic modules. K\"othe studied this class of rings which is an important subclass of representation-finite rings. He showed that commutative artinian rings which have this property are serial. Also he posed the question to classify the non-commutative rings with this property \cite{K} (see also \cite{Jain} and references therein). Kawada completely solved the K\"othe's problem for the basic finite dimensional $K$-algebras. Kawada's papers contain a set of 19 conditions which characterize Kawada algebras, as well as, the list of all possible finitely generated indecomposable modules \cite{Kawada1, Kawada2, Kawada3}.  A ring $\Lambda$ is called {\it left Kawada} if any ring Morita equivalent to $\Lambda$ is a left K\"othe ring \cite{r}. Ringel showed that any finite dimensional $K$-algebra of finite representation type is Morita equivalent to a K\"othe algebra. By using of the multiplicity-free of top and soc of finitely generated indecomposable modules, he also gave a characterization of Kawada algebras \cite{r}. In \cite{za} the authors proved the Ringel's result for artinian rings.

Since any left K\"othe ring is representation-finite, the above results of Auslander propose to use the Auslander ring and the functor category Mod$((\Lambda$-mod$)^{\rm op})$ for study left K\"othe rings. In this paper we study left K\"othe rings by using their Auslander rings and their functor categories.
We say that a semiperfect ring with enough idempotents $R$ is {\it generalized left {\rm (}resp., right{\rm )} QF-2} if every indecomposable projective unitary left (resp., right) $R$-module $P$ has multiplicity-free socle (i.e. the composition factors of soc$(P)$ are pairwise non-isomorphic). In the following theorem, by using the notion of generalized right QF-2 rings, we give a characterization of basic left K\"othe rings.

\newtheorem{thm}{Theorem}
\begin{theorem a*}$($See Theorem \ref{r33}$)$
Let $\Lambda$ be a basic ring. Then the following conditions are equivalent.
\begin{itemize}
\item[$(a)$] $\Lambda$ is a left K\"othe ring.
\item[$(b)$] $\Lambda$ is a ring of finite representation type and the Auslander ring $T$ of $\Lambda$ is a generalized right QF-2 ring.
\item[$(c)$] $\Lambda$ is a left artinian ring and $T=\widehat{{\rm End}}_{\Lambda}(V)$ is a left locally finite generalized right QF-2 ring, where $V$ is a direct sum of modules in a complete set of representative of the isomorphic classes of  finitely generated indecomposable left $\Lambda$-modules.
\item[$(d)$] $\Lambda$ is a left artinian ring and for each indecomposable left $\Lambda$-module $M$, ${\rm Hom}_{\Lambda}(M,-) \simeq \varphi(-)$ for some positive-primitive formula $\varphi(x)$ over $\Lambda$.
\item[$(e)$] $\Lambda$ is a left artinian ring and every indecomposable projective object in  Mod$((\Lambda$-mod$)^{\rm op})$ has finitely generated essential multiplicity-free socle.
\end{itemize}
\end{theorem a*}

Using the above theorem, we prove the following theorem, which gives a characterization of left Kawada rings.

\begin{theorem b*}$($See Theorem \ref{ka}$)$
There exists a bijection between the Morita equivalence classes of left Kawada rings and the Morita equivalence classes of Auslander generalized right QF-2 rings.
\end{theorem b*}

Before proving our main results in Section 3 we prove some preliminary results in the following section.
\subsection{Notation }
Throughout this paper all rings are associative with unit unless otherwise stated. Let $R$ be a ring (not necessary with unit). We write all homomorphisms of left (resp., right) $R$-modules on the right (resp., left), so $fg$ (resp., $g \circ f$) means ``first $f$ then $g$".  We denote by $R$-Mod (resp., Mod-$R$) the category of all left (resp., right) $R$-modules and by $J(R)$ the Jacobson radical of $R$.  Also we denote by $R$-mod (resp., mod-$R$) the category of all finitely generated left (resp., right) $R$-modules. A left (resp., right) $R$-module $M$ is called {\it unitary} if $RM=M$ (resp., $MR=M$). We denote by $R$Mod (resp., Mod$R$) the category of all unitary left (resp., right) $R$-modules. We denote by Mod$((R$-mod$)^{\rm op})$ the category of all additive covariant functors from $R$-mod to the category of abelian groups, which is an abelian category. We write $\bigoplus_A M$ and $\prod_A M$ for the direct sum of card$(A)$ copies of an $R$-module $M$ and the direct product of card$(A)$ copies of an $R$-module $M$, respectively. For a left $R$-module $V$ we denoted by ${\rm Add}(V)$ the full subcategory of $R$-Mod whose objects are all left $R$-modules that are isomorphic to direct summands of $\bigoplus_A V$ for any set $A$. We denoted by ${\rm Proj}(R)$ (resp., ${\rm Proj}(R^{\rm op})$) the full subcategory of $R$Mod (resp., Mod$R$) whose objects are projective left (resp., right) $R$-modules. Let $N$ be a unitary left $R$-module. We denote by $E(N)$, ${\rm rad}(N)$, ${\rm top}(N)$ and ${\rm soc}(N)$ the injective hull of $N$ in $R$Mod, radical of $N$, top of $N$ and socle of $N$, respectively. Let $f : X \rightarrow Y$ be an $R$-module homomorphism and assume that $M$ is an $R$-submodule of $X$. We denote by $f\mid_M$ the restriction of $f$ to $M$. Let  $L$ and $N$ be two left $R$-modules. The submodule ${\rm Re}(L, N)=\bigcap \lbrace {\rm Ker}f~|~ f\in {\rm Hom}_{R}(L,N) \rbrace$ is called {\it reject of $N$ in $L$}.

\section{Preliminaries}

 Let $\mathcal{U}$ be a non-empty set of left $\Lambda$-modules. A left $\Lambda$-module $M$ is said to be {\it generated} by $\mathcal{U}$ if for every pair of distinct morphisms $f,g : M \rightarrow B$ in $\Lambda$-Mod there exists a morphism $h : U \rightarrow M$ with $U \in \mathcal{U}$ such that  $h f \neq hg$. Also, $\mathcal{U}$ is called a {\it generating set} for $\Lambda$-Mod if every left $\Lambda$-module generated by $\mathcal{U}$. A left $\Lambda$-module $U$ is called {\it generator} in $\Lambda$-Mod if set $\mathcal{U}=\lbrace U \rbrace$  is a generating set for $\Lambda$-Mod (see \cite[Ch. V, Sect. 7]{s}). Let $\Lambda$ be a ring of finite representation type and $\lbrace X_1, \cdots, X_n\rbrace$ be a complete set of representative of the isomorphic classes of finitely generated indecomposable left $\Lambda$-modules. We recall that $X=\bigoplus_{i=1}^nX_i$ is called an Auslander generator of $\Lambda$-mod. Also the endomorphism ring of $X$ is called {\it Auslander ring of} $\Lambda$. Note that by \cite[Proposition 3.6]{ausla2}, the Auslander ring of $\Lambda$ is an artinian ring. \\

Let $\Lambda$ be a ring and $\lbrace V_{\alpha}~|~ \alpha \in J \rbrace$ be a family of finitely generated left $\Lambda$-modules. Set $V=\bigoplus_{\alpha \in J} V_{\alpha}$ and for each $\alpha \in J$, letting $e_{\alpha}=\pi_{\alpha}\varepsilon_{\alpha}$, where $\pi_{\alpha}:V \rightarrow V_{\alpha}$ is the canonical projection and $\varepsilon_{\alpha}:V_{\alpha} \rightarrow V$ is the canonical injection. For each left $\Lambda$-module $X$, we define as in \cite[Page 40]{fu}, $\widehat{{\rm{Hom}}}_{\Lambda}(V,X)=\lbrace f \in {\rm Hom}_{\Lambda}(V,X)~|~(V)e_{\alpha} f=0~ {\rm for ~almost~ all}~ \alpha \in J \rbrace$. For $X=V$, we write $\widehat{{\rm{Hom}}}_{\Lambda}(V,V)=\widehat{{\rm End}}_{\Lambda}(V)$.  Let $R$ be a ring (not necessary with unit). $R$ is called a {\it ring with enough idempotents} if there exists a family $\lbrace q_{\alpha}~|~\alpha \in I \rbrace$ of pairwise orthogonal idempotents of $R$ with $R=\bigoplus_{\alpha \in I}Rq_{\alpha}=\bigoplus_{\alpha \in I}q_{\alpha}R$ (see \cite[Page 39]{fu}).  $T=\widehat{{\rm End}}_{\Lambda}(V)$ is a ring with enough idempotents because of $T=\bigoplus_{\alpha \in J}Te_{\alpha}=\bigoplus_{\alpha \in J}e_{\alpha}T$. Fuller in \cite[Page 40]{fu} defined a covariant functor $\widehat{{\rm{Hom}}}_{\Lambda}(V,-):\Lambda$-Mod$\rightarrow T$Mod as follows. For any morphism $f: X \rightarrow Y$ in $\Lambda$-Mod, he defined $\widehat{{\rm Hom}}_{\Lambda}(V,f): \widehat{{\rm Hom}}_{\Lambda}(V,X) \rightarrow \widehat{{\rm Hom}}_{\Lambda}(V,Y)$ via $g \mapsto gf$. From \cite[Pages 40-41]{fu} we observe that the covariant functor $\widehat{{\rm{Hom}}}_{\Lambda}(V,-)$ is a left exact functor and also preserves direct sums.  Moreover it is an additive equivalence between the full subcategory ${\rm Add}(V)$ of $\Lambda$-Mod and the full subcategory Proj$(T)$ of $T$Mod with the inverse equivalence $V \otimes_T -$.  Note that the covariant functor $\widehat{{\rm{Hom}}}_{\Lambda}(V,-):\Lambda$-Mod$\rightarrow T$Mod preserves indecomposable modules, injective modules and essential extensions when $V$ is a generator in $\Lambda$-Mod (see \cite[Proposition 51.7]{wi}). Also it is easily seen that $V$ is a generator in $\Lambda$-Mod when $\Lambda$ is left artinian and $\lbrace V_{\alpha}~|~ \alpha \in J \rbrace$ is a complete set of representative of the isomorphic classes of finitely generated indecomposable left $\Lambda$-modules. By using \cite[Proposition 4.2 and Corollary 4.8]{ausla2} and \cite[Proposition 46.7]{wi} we have the following result.

\begin{Lem}\label{p0}
Let $\Lambda$ be a ring of finite representation type, $X$ be an Auslander generator and $T={\rm End}_{\Lambda}(X)$. Then
\begin{itemize}
\item[$(a)$] The functor ${\rm Hom}_{\Lambda}(X,-): \Lambda$-{\rm Mod}$ \rightarrow${\rm Proj}$(T)$ is an equivalence which preserves and reflects finitely generated modules.
\item[$(b)$] The functor ${\rm Hom}_{\Lambda}(X,-)$ preserves essential extension.
\item[$(c)$] If $E$ is an injective left $\Lambda$-module, then ${\rm Hom}_{\Lambda}(X,E)$ is an injective left $T$-module.
\end{itemize}
\end{Lem}

A ring $\Lambda$ is called {\it semiperfect} if $\Lambda/J(\Lambda)$ is a semisimple ring and idempotents in $\Lambda/J(\Lambda)$ can be lifted to $\Lambda$ (see \cite[Page 303]{an}). It is well-known that a ring with enough idempotents $R$ is semiperfect if and only if every finitely generated unitary left $R$-module has a projective cover in $R$Mod if and only if every finitely generated projective unitary left $R$-module is a direct sum of local modules (see \cite[Page 95]{f} and \cite[Proposition 49.10]{wi}).\\

A semiperfect ring $\Lambda$ is called {\it left {\rm (}resp., right{\rm )} QF-2} if every indecomposable projective left (resp., right) $\Lambda$-module has a simple essential socle (see \cite[Sect. 4]{f}).  A left $\Lambda$-module $M$ has {\it multiplicity-free socle} if composition factors of soc$(M)$ are pairwise non-isomorphic (see \cite[Sect. 1]{r}).  We say that a semiperfect ring with enough idempotents $R$ is  {\it generalized left {\rm (}resp., right{\rm )} QF-2} if every indecomposable projective unitary left (resp., right) $R$-module has multiplicity-free socle.

\begin{Pro}\label{r1}
Let $\Lambda$ be a ring of finite representation type.  Then the following conditions are equivalent.
\begin{itemize}
\item[$(a)$] The Auslander ring of $\Lambda$ is a generalized left QF-2 ring.
\item[$(b)$] Every indecomposable left $\Lambda$-module has multiplicity-free socle.
\end{itemize}
\end{Pro}
\begin{proof}
Let $X$ be an Auslander generator of $\Lambda$-mod and $T = {\rm End}_{\Lambda}(X)$. Assume that $M$ is a finitely generated left $\Lambda$-module. Since $\Lambda$ is left artinian, by \cite[Propositions 21.3 and 31.4]{wi}, soc$(M)$ is a finitely generated and essential submodule of $M$. It is easy to see that soc$(M)=\bigoplus_{i=1}^nT_i$, where $n \in {\Bbb{N}}$ and each $T_i$ is a simple left $\Lambda$-module. Since soc$(M)$ is an essential submodule of $M$,  we get $E(M) \cong \bigoplus_{i=1}^n E(T_i)$ as $\Lambda$-modules. On the other hand, by Lemma \ref{p0}(b), there exists an essential monomorphism  ${\rm Hom}_{\Lambda}(X,M) \rightarrow {\rm Hom}_{\Lambda}(X,E(M))$ as $T$-modules. Also by Lemma \ref{p0}(c), we see that ${\rm Hom}_{\Lambda}(X,E(M))$ is an injective left $T$-module. Therefore $$E({\rm Hom}_{\Lambda}(X,M)) \cong {\rm Hom}_{\Lambda}(X,E(M)) \cong \bigoplus_{i=1}^n {\rm Hom}_{\Lambda}(X,E(T_i))$$ as $T$-modules. ${\rm End}_{T}({\rm Hom}_{\Lambda}(X,E(T_i)))$ is a local ring because the functor
\begin{center}
${\rm Hom}_{\Lambda}(X,-):\Lambda$-Mod$ \rightarrow$Proj$(T)$
\end{center}
is an equivalence and ${\rm End}_{\Lambda}(E(T_i))$ is a local ring. \\
$(a)\Rightarrow (b)$. Let $M$ be an indecomposable left $\Lambda$-module.  Since $\Lambda$ is a ring of finite representation type, by \cite[Corollary 4.8]{ausla2}, $M$ is finitely generated. Hence soc$(M)=\bigoplus_{i=1}^nT_i$, where $n \in {\Bbb{N}}$ and each $T_i$ is a simple left $\Lambda$-module. We show that $T_i \ncong T_j$ when $i \neq j$.  By Lemma \ref{p0}(a), ${\rm Hom}_{\Lambda}(X,M)$ is a finitely generated indecomposable projective left $T$-module. Since $T$ is an artinian ring,  ${\rm soc}({\rm Hom}_{\Lambda}(X,M))= \bigoplus_{i=1}^tS_i$, where $t \in {\Bbb{N}}$, each $S_i$ is a simple left $T$-module.  Moreover  $E({\rm Hom}_{\Lambda}(X,M)) \cong \bigoplus_{i=1}^tE(S_i)$ as $T$-modules. By assumption, $S_i \ncong S_j$ when $i \neq j$. On the other hand, by above statement $E({\rm Hom}_{\Lambda}(X,M)) \cong \bigoplus_{i=1}^n {\rm Hom}_{\Lambda}(X,E(T_i))$ as $T$-modules, where $n \in {\Bbb{N}}$, each $T_i$ is a simple left $\Lambda$-module and soc$(M)=\bigoplus_{i=1}^nT_i$. Also ${\rm End}_{T}({\rm Hom}_{\Lambda}(X,E(T_i)))$ is a local ring for each $1 \leq i \leq n$. Therefore  $t=n$ and for each $1 \leq i \leq n$, $E(S_i) \cong {\rm Hom}_{\Lambda}(X,E(T_i))$ as $T$-modules. We can certainly conclude that  $T_i \ncong T_j$ when $i \neq j$ since otherwise  $E(T_i) \cong E(T_j)$ as $\Lambda$-modules for some $i \neq j$. This implies that ${\rm Hom}_{\Lambda}(X,E(T_i)) \cong {\rm Hom}_{\Lambda}(X,E(T_j))$ as $T$-modules. $E(S_i) \cong E(S_j)$ yields $S_i \cong S_j$ which is a contradiction.  Therefore every indecomposable left $\Lambda$-module has multiplicity-free socle.\\
$(b)\Rightarrow (a)$. Let $P$ be an indecomposable projective left $T$-module. Then $P$ is a finitely generated left $T$-module because $T$ is an artinian ring.  Hence by Lemma \ref{p0}(a), $P \cong {\rm Hom}_{\Lambda}(X,M)$ for some a finitely generated indecomposable left $\Lambda$-module $M$. Then $E(P) \cong E({\rm Hom}_{\Lambda}(X,M)) \cong \bigoplus_{i=1}^n {\rm Hom}_{\Lambda}(X,E(T_i))$ and ${\rm End}_{T}({\rm Hom}_{\Lambda}(X,E(T_i)))$ is a local ring for each $1 \leq i \leq n$, where $n \in {\Bbb{N}}$, ${\rm soc}(M)=\bigoplus_{i=1}^n T_i$ and each $T_i$ is a simple left $\Lambda$-module. On the other hand, ${\rm soc}(P) = \bigoplus_{i=1}^s S_i$ and $E(P) \cong \bigoplus_{i=1}^s E(S_i)$, where $s \in {\Bbb{N}}$ and each $S_i$ is a simple left $T$-module. Therefore $s=n$ and $E(S_i) \cong {\rm Hom}_{\Lambda}(X,E(T_i))$ for each $1 \leq i \leq n$. If $S_i \cong S_j$ as $T$-modules for some $i \neq j$, then $E(S_i) \cong E(S_j)$ as $T$-modules and hence  ${\rm Hom}_{\Lambda}(X,E(T_i)) \cong {\rm Hom}_{\Lambda}(X,E(T_j))$ as $T$-modules. So by Lemma \ref{p0}(a), we have $E(T_i) \cong E(T_j)$ as $\Lambda$-modules. It follows that $T_i \cong T_j$ as $\Lambda$-modules which is a contradiction. Consequently $T$ is a generalized left QF-2 ring and the result follows.
\end{proof}

Remark \ref{se} and Lemma \ref{l1} are well-known to experts, but since we could not find a suitable reference in the literature, we include these results here.

\begin{Rem}\label{se}
{\rm  Let $\Lambda$ be a left artinian ring and $\lbrace V_{\alpha}~|~\alpha \in J \rbrace$ be a family of finitely generated indecomposable left $\Lambda$-modules. Set $V=\bigoplus_{\alpha \in J}V_{\alpha}$. Let $\pi_{\alpha}: V \rightarrow V_{\alpha}$ be the canonical projection and $\varepsilon_{\alpha}: V_{\alpha} \rightarrow V$ be the canonical injection. Set $e_{\alpha}=\pi_{\alpha}\varepsilon_{\alpha}$ for each $\alpha \in J$. Then $T=\widehat{{\rm End}}_{\Lambda}(V) =\bigoplus_{\alpha \in J}Te_{\alpha}=\bigoplus_{\alpha \in J}e_{\alpha}T$. Since $\widehat{{\rm{Hom}}}_{\Lambda}(V,-): {\rm Add}(V) \rightarrow {\rm Proj}(T)$ is an equivalence of categories and $Te_{\alpha} \cong \widehat{{\rm Hom}}_{\Lambda}(V, V_{\alpha})$ as $T$-modules we have for each $\alpha \in J$,  $e_{\alpha}Te_{\alpha} \cong {\rm End}_{\Lambda}(V_{\alpha})$ as rings.  $V_{\alpha}$ has finite length and also it is indecomposable, then by \cite[Proposition 32.4(3)]{wi}, $e_{\alpha}Te_{\alpha}$ is a local ring. It follows that every finitely generated projective unitary left $T$-module is a direct sum of local modules. Consequently $T$ is a semiperfect ring.}
\end{Rem}

\begin{Lem}\label{l1}
Let $\Lambda$ be a ring and assume that $\lbrace V_{\alpha}~|~\alpha \in I \rbrace$ is a family of non-isomorphic finitely generated indecomposable left $\Lambda$-modules. Set $T=\widehat{{\rm End}}_{\Lambda}(V)$, where $V=\bigoplus_{\alpha \in I}V_{\alpha}$. If $\Lambda$ is a left artinian ring, then  $\lbrace Te_{\alpha}/{\rm rad}(Te_{\alpha})~|~\alpha \in I \rbrace$ is a complete set of representative of the isomorphic classes of  simple unitary left $T$-modules, where $e_{\alpha}=\pi_{\alpha}\varepsilon_{\alpha}$, $\pi_{\alpha}: V \rightarrow V_{\alpha}$ is the canonical projection and $\varepsilon_{\alpha}: V_{\alpha} \rightarrow V$ is the canonical injection.
\end{Lem}
\begin{proof}
By Remark \ref{se}, ${\rm End}_T(Te_{\alpha})$ is a local ring.  $Te_{\alpha}$ is a projective unitary left $T$-module and hence ${\rm rad}(Te_{\alpha})$ is a superfluous maximal submodule of $Te_{\alpha}$. Then $Te_{\alpha}/{\rm rad}(Te_{\alpha})$ is a simple unitary left $T$-module. Also the canonical projection $Te_{\alpha} \rightarrow Te_{\alpha}/{\rm rad}(Te_{\alpha})$ is a projective cover of $Te_{\alpha}/{\rm rad}(Te_{\alpha})$ in $T$Mod. If $Te_{\alpha}/{\rm rad}(Te_{\alpha}) \cong Te_{\beta}/{\rm rad}(Te_{\beta})$ as $T$-modules, then $Te_{\alpha} \cong Te_{\beta}$. It is easy to see that $V_{\alpha} \cong V_{\beta}$ as $\Lambda$-modules and hence $\alpha=\beta$. Therefore $\lbrace Te_{\alpha}/{\rm rad}(Te_{\alpha})~|~\alpha \in I \rbrace$ is a set of non-isomorphic simple unitary left $T$-modules. It remains to prove that  every simple unitary left $T$-module $S$ is isomorphism to $Te_{\alpha}/{\rm rad}(Te_{\alpha})$ for some $\alpha \in I$. Let $S$ be a simple unitary left $T$-module. By Remark \ref{se}, $T$ is a semiperfect ring and hence $S$ has a projective cover in $T$Mod.  Let $f: P \rightarrow S$ be a projective cover of $S$ in $T$Mod. Since ${\rm Ker}f$ is a maximal superfluous submodule of $P$, $P$ is an indecomposable unitary left $T$-module. It follows that $P \cong Te_{\alpha}$ as $T$-modules for some $\alpha \in I$ and hence $S \cong Te_{\alpha}/{\rm rad}(Te_{\alpha})$ as $T$-modules and the result follows.
\end{proof}

Let $\Lambda$ be a ring and $\lbrace V_{\alpha}~|~ \alpha \in J \rbrace$ be a family of finitely generated left $\Lambda$-modules. Set $V=\bigoplus_{\alpha \in J} V_{\alpha}$ and $R=\widehat{{\rm End}}_{\Lambda}(V)$. As in \cite[Page 40]{fu}, we call $R$  {\it the left functor ring of $\Lambda$} when $\lbrace V_{\alpha}~|~ \alpha \in J \rbrace$ is a complete set of representative of the isomorphic classes of finitely generated left $\Lambda$-modules. Two rings  with enough idempotents $R$ and $S$ are said to be {\it Morita equivalent} if there exists an additive covariant equivalence between the category of unitary left $R$-modules and the category of unitary left $S$-modules (see \cite[Sect. 3]{a}).

\begin{Pro}\label{r3}
Let $\Lambda$ be a ring and $\lbrace V_{\alpha}~|~ \alpha \in J \rbrace$ be a complete set of representative of the isomorphic classes of finitely generated indecomposable left $\Lambda$-modules. Set $V=\bigoplus_{\alpha \in J} V_{\alpha}$ and $T=\widehat{{\rm End}}_{\Lambda}(V)$. If every left $\Lambda$-module is a direct sum of finitely generated modules, then $T$ is Morita equivalent to the left functor ring $R$ of $\Lambda$.
\end{Pro}
\begin{proof}
First we show that ${\rm Proj}(R)$ and ${\rm Proj}(T)$ are equivalent. Let $\lbrace U_{\alpha}~|~ \alpha \in I \rbrace$ be a complete set of representative of the isomorphic classes of finitely generated left $\Lambda$-modules such that $R=\widehat{{\rm End}}_{\Lambda}(U)$, where $U= \bigoplus_{\alpha \in I}U_{\alpha}$.  By assumption  Add$(U)=\Lambda$-Mod and also by  \cite[Theorem 4.4]{chase}, we see that $\Lambda$ is a left artinian ring. Since  Add$(U)=\Lambda$-Mod,  the functor  $\widehat{{\rm Hom}}_{\Lambda}(U,-): \Lambda$-Mod$ \rightarrow {\rm Proj}(R)$ is an additive equivalence with the inverse equivalence $U\otimes_R-$. On the other hand,  every finitely generated left $\Lambda$-module is a direct sum of indecomposable modules because $\Lambda$ is left artinian. Then Add$(V)=\Lambda$-Mod and we have an additive equivalence $\widehat{{\rm Hom}}_{\Lambda}(V,-): \Lambda$-Mod$ \rightarrow {\rm Proj}(T)$ with the inverse equivalence $V\otimes_T-$. Therefore we obtain an additive equivalence $\widehat{{\rm Hom}}_{\Lambda}(V,U\otimes_R-): {\rm Proj}(R) \rightarrow {\rm Proj}(T)$ with the inverse equivalence  $\widehat{{\rm Hom}}_{\Lambda}(U,V\otimes_T-): {\rm Proj}(T) \rightarrow {\rm Proj}(R)$. Now we show that this equivalence preserves finitely generated modules. Let $Y$ be a finitely generated projective unitary left $R$-module. Set $X=U\otimes_RY$. Since $\widehat{{\rm Hom}}_{\Lambda}(U,X) \cong Y$ and $\widehat{{\rm Hom}}_{\Lambda}(U,-)$ preserves direct sums, $X$ is a finitely generated left $\Lambda$-module and hence $X$ is a finite direct sum of finitely generated indecomposable left $\Lambda$-modules. Then $\widehat{{\rm Hom}}_{\Lambda}(V,X)$ is a finitely generated projective unitary left $T$-module. Consequently the equivalence $\widehat{{\rm Hom}}_{\Lambda}(V,U\otimes_R-): {\rm Proj}(R) \rightarrow {\rm Proj}(T)$  preserves finitely generated projective modules. By the similar argument, we can see that it also reflects finitely generated projective modules. Therefore the result follows from \cite[Theorem 3.10]{zn}.
\end{proof}

Let $\Lambda$ be a ring and $\mathcal{U}$ be a non-empty set of left $\Lambda$-modules. A left $\Lambda$-module $M$ is said to be {\it cogenerated by} $\mathcal{U}$ if for every pair of distinct morphisms $f,g : B \rightarrow M$ in $\Lambda$-Mod, there exists a morphism $h : M \rightarrow U$ with $U \in \mathcal{U}$ such that $fh \neq gh$. Also, $\mathcal{U}$ is called a {\it cogenerating set} for $\Lambda$-Mod if every left $\Lambda$-module cogenerated by $\mathcal{U}$. A left $\Lambda$-module $U$ is called {\it cogenerator} if the set $\mathcal{U}=\lbrace U \rbrace$ is a cogenerating set for $\Lambda$-Mod (see \cite[Ch. V, Sect. 7]{s}). A ring $\Lambda$ is called {\it left Morita} if there is an injective cogenerator left $\Lambda$-module $Q$ such that it is an injective cogenerator right $\Delta$-module and $\Lambda \cong {\rm End}_{\Delta}(Q)$, where $\Delta = {\rm End}_{\Lambda}(Q)$ (see \cite[Ch. 9, Sect. 47]{wi}). In this case we say that $\Lambda$ is left Morita to $\Delta$.

\begin{Pro}\label{p6}
Let $\Lambda$ be a ring and $R$ be the left functor ring of $\Lambda$. If $\Lambda$ is a left artinian ring and every finitely generated indecomposable projective unitary right $R$-module has finitely generated essential socle, then $\Lambda$ is a left Morita ring.
\end{Pro}
\begin{proof}
By \cite[Theorem 3.3]{f} it is enough to show that ${\rm soc}(R_R)$ is essential in $R_R$ and also $R_R$ has only finitely many non-isomorphic simple right $R$-modules. Suppose that $\lbrace U_{\alpha}~|~\alpha \in I \rbrace$ is a complete set of representative of the isomorphic classes of finitely generated left $\Lambda$-modules and $R=\widehat{{\rm End}}_{\Lambda}(U)$, where $U=\bigoplus_{\alpha \in I}U_{\alpha}$. $U$ is a generator in $\Lambda$-Mod. Since $\Lambda$ is left artinian, by \cite[Proposition 51.7(9)]{wi}, $R$ is a semiperfect ring. Therefore  every finitely generated projective unitary right $R$-module is a direct sum of indecomposable modules. Hence ${\rm soc}(R_R)$ is an essential submodule of $R_R$. Now we show that $R_R$ has only finitely many non-isomorphic simple right $R$-modules. $_{\Lambda}\Lambda \cong U_{\alpha}$ for some $\alpha \in I$ and so  $U \cong e_{\alpha}R$ as $R$-modules, where $e_{\alpha}=\pi_{\alpha}\varepsilon_{\alpha}$, $\pi_{\alpha}:U \rightarrow U_{\alpha}$ is the canonical projection and $\varepsilon_{\alpha}:U_{\alpha} \rightarrow U$ is the canonical injection. Moreover $e_{\alpha}R = \bigoplus_{i=1}^{t_{\alpha}} P_i$, where each $P_i$ is an indecomposable right $R$-module and hence ${\rm soc}(e_{\alpha}R) \cong \bigoplus_{i=1}^{t_{\alpha}}{\rm soc}(P_i)$ as $R$-modules. Since every finitely generated indecomposable projective unitary right $R$-module has finitely generated socle, ${\rm soc}(e_{\alpha}R)$ is finitely generated. In order to complete the proof, it is sufficient to show that for each simple right $R$-submodule $L$ of $R$, there is an $R$-module monomorphism $L \rightarrow e_{\alpha}R$. Since $e_{\alpha}R$ is faithful,  ${\rm Re}(R_R, e_{\alpha}R)=0$.  It follows that by \cite[Proposition 14.3]{wi}, there exists an $R$-module monomorphism $\varphi: R_R \rightarrow \prod_A e_{\alpha}R$.  Let $L$ be a simple submodule of $R_R$. Then $L=aR$ for some $0 \neq a \in R$. Put $\varphi(a)=(y_i)$, where each $y_i \in e_{\alpha}R$. Since $\varphi$ is a monomorphism,  $y_i \neq 0$ for some $i$. Without loss of generality we can assume that $y_1 \neq 0$. Since $R$ is a ring with enough idempotents, there exists an idempotent $e \in R$ such that $ae=a$ and $y_1e=y_1$. Therefore the $R$-module homomorphism $\pi_1 \circ \overline{\varphi}: L \rightarrow e_{\alpha}R$ is a non-zero morphism, where $\overline{\varphi}:= \varphi|_{L}$ and $\pi_1: \prod_A e_{\alpha}R \rightarrow e_{\alpha}R$ is the canonical projection and the result follows.
\end{proof}

A left $\Lambda$-module $W$ is called {\it minimal cogenerator} if $W \cong \bigoplus_{i\in I} E(S_i)$, where $\lbrace S_i~|~i\in I\rbrace$ is a complete set of representative of the isomorphic classes of simple left $\Lambda$-modules (see \cite[Ch. 3, Sect. 17]{wi}). Let $\Lambda$ and $\Delta$ be two rings. An additive contravariant functor $F : \Lambda$-mod $\rightarrow$ mod-$\Delta$ is called duality if it is an equivalence of categories (see \cite[Ch. 9, Sect. 47]{wi}).

\begin{Rem}\label{r2}{\rm
Let $\Lambda$ be a ring and assume that $W$ is a minimal cogenerator left $\Lambda$-module. Assume that $\Lambda$ is left artinian and $W$ is a finitely generated left $\Lambda$-module. Then we derive from \cite[Proposition 47.15]{wi} that $\Delta = {\rm End}_{\Lambda}(W)$ is a right artinian ring, $W$ is an injective cogenerator in Mod-$\Delta$ and $\Lambda \cong {\rm End}_{\Delta}(W)$ as rings. Therefore $\Lambda$ is a left Morita to $\Delta$. Also we conclude from \cite[Proposition 47.3]{wi} that the functor ${\rm Hom}_{\Lambda}(-,W): \Lambda$-mod$\rightarrow$ mod-$\Delta$ is a duality with the inverse duality ${\rm Hom}_{\Delta}(-,W):$ mod-$\Delta \rightarrow \Lambda$-mod. By \cite[Proposition 24.5]{an} for each finitely generated left $\Lambda$-module $X$, the lattice of all $\Lambda$-submodules of $M$ and the lattice of all $\Delta$-submodules of ${\rm Hom}_{\Lambda}(M,W)$ are anti-isomorphic. Hence for each finitely generated left $\Lambda$-module $X$, soc$({\rm Hom}_{\Lambda}(X,W)) \cong {\rm Hom}_{\Lambda}({\rm top}(X),W)$ as $\Delta$-modules.}
\end{Rem}

\section{Main results}

Let $A$ and $B$ be two $n \times 1$ and $n \times m$ matrices over $\Lambda$, respectively. A formula $\varphi(x)$ is called {\it left positive-primitive formula} over $\Lambda$ in the free variable $x$ if it has the form
$\exists \mathbf{y} ~~\begin{pmatrix}
 A & B
\end{pmatrix}(x, \mathbf{y})^t=0$, where $\mathbf{y}= (c_1, \cdots ,c_m)$ with entries in $\Lambda$ (see \cite{pe}). A left positive-primitive formula is completely determined by the matrices $A$ and $B$.  For a left $\Lambda$-module $M$, the above left positive-primitive formula define a subgroup $$\varphi(M)=\lbrace a \in M~|~\exists c_1,\cdots,c_m ~{\rm in}~M ~{\rm with}~\begin{pmatrix}
 A & B
\end{pmatrix}{\begin{pmatrix}
a & c_1 & \cdots &c_m
\end{pmatrix}}^t=0\rbrace.$$
If $f: M \rightarrow N$ is a $\Lambda$-module homomorphism, then $(a)f \in \varphi(N)$ when $a \in \varphi(M)$. This yields a covariant functor $\varphi(-): \Lambda$-{\rm mod}$ \rightarrow {\rm Ab}$ which is a subfunctor of the forgetful functor ${\rm Hom}_{\Lambda}({_\Lambda\Lambda},-)$. Note that a subfunctors $F$ of the forgetful functor ${\rm Hom}_{\Lambda}({_\Lambda\Lambda},-)$ is finitely generated if and only if $F \simeq \varphi(-)$ for some positive-primitive formula $\varphi(x)$ over $\Lambda$ (see \cite[Corollary 12.4]{pe}).

\begin{Rem}\label{r22}
{\rm Let $\Lambda$ be a ring and suppose that $R$ is the left functor ring of $\Lambda$. Then $R=\widehat{{\rm End}}_{\Lambda}(U)$, where $U= \bigoplus_{\alpha \in I}U_{\alpha}$ and $\lbrace U_{\alpha}~|~ \alpha \in I \rbrace$ is a complete set of representative of the isomorphic classes of finitely generated left $\Lambda$-modules. We define as in \cite{wi} a functor
\begin{center}
$\textbf{F}: $ {\rm Mod}$R \longrightarrow {\rm Mod}((\Lambda$-mod$)^{\rm op})$
\end{center}
as follows. For each $M \in$ Mod$R$, we set $\textbf{F}(M) = {\rm Hom}_R({\rm Hom}_{\Lambda}(-,U),M)$. It is easy to check that ${\rm Hom}_R({\rm Hom}_{\Lambda}(-,U),M):\Lambda$-mod$ \rightarrow {\rm Ab}$ is an additive covariant functor. If $f: M \rightarrow N$ is a morphism in Mod$R$, then $\textbf{F}(f) = {\rm Hom}_R({\rm Hom}_{\Lambda}(-,U),f)$ which is a natural transformation from ${\rm Hom}_R({\rm Hom}_{\Lambda}(-,U),M)$ to ${\rm Hom}_R({\rm Hom}_{\Lambda}(-,U),N)$. By \cite[Proposition 52.5]{wi}, $\textbf{F}$ is an equivalence of categories.  It is not difficult to see that the functor $\textbf{F}$ preserves and reflects simple objects, projective objects, monomorphisms, epimorphisms, essential monomorphisms and direct sums. Let $F$ be an object in Mod$((\Lambda$-mod$)^{\rm op})$. There exists a subfunctor $G$ of $F$ such that the following diagram is commutative
\begin{displaymath}
\xymatrix{
\mathbf{F}({\rm soc}(X)) \ar[dr]_{h} \ar@{->}[r]^{\mathbf{F}(\ell)} & \mathbf{F}(X) \ar[r]^{\varphi} & F  \\
&  G \ar@{^{(}->}[ur]\\
 }
\end{displaymath}
, where $\varphi$ and $h$ are isomorphisms and $\ell: {\rm soc}(X) \rightarrow X$ is the canonical injection in Mod$R$. Note that the functor $G$ is independent of choice $X$. The subfunctor $G$ of $F$ is called {\rm socle of $F$} and we denote it by ${\rm soc}(F)$.  The indecomposable projective objects with finitely generated essential multiplicity-free socle in Mod$((\Lambda$-mod$)^{\rm op})$ correspond to the indecomposable projective unitary right $R$-modules with finitely generated essential multiplicity-free socle under the equivalence $\textbf{F}$. From now on, we fix the functor $\textbf{F}$.}
\end{Rem}

A semiperfect ring $\Lambda$ is called basic if $\Lambda/J(\Lambda)$ is a direct sum of division rings \cite{an}. By \cite[Proposition 27.14]{an},  any semiperfect ring is Morita equivalent to a basic ring. A left $\Lambda$-module $M$ has {\it multiplicity-free top} if composition factors of top$(M)$ are pairwise non-isomorphic (see \cite[Sect. 1]{r}). Let $R$ be a ring with enough idempotents. We recall that $R$ is called {\it left locally noetherian {\rm (}resp., left locally artinian{\rm )}} if every finitely generated unitary left $R$-module is noetherian (resp., artinian) (see \cite[Ch. 5, Sect. 27]{wi}). Also $R$ is called {\it left locally finite} if it is both left locally artinian and left locally noetherian (see \cite[Ch. 6, Sect. 32]{wi}).  \\

Note that $R$ being left (resp., right) artinian generalized left (resp., right) QF-2 is a Morita invariant property. We are now in a position to prove our main result.

\begin{The}\label{r33}
Let $\Lambda$ be a basic ring. Then the following conditions are equivalent.
\begin{itemize}
\item[$(a)$] $\Lambda$ is a left K\"othe ring.
\item[$(b)$] $\Lambda$ is a ring of finite representation type and the Auslander ring $T$ of $\Lambda$ is a generalized right QF-2 ring.
\item[$(c)$] $\Lambda$ is a left artinian ring and $T=\widehat{{\rm End}}_{\Lambda}(V)$ is a left locally finite generalized right QF-2 ring, where $V$ is a direct sum of modules in a complete set of representative of the isomorphic classes of  finitely generated indecomposable left $\Lambda$-modules.
\item[$(d)$] $\Lambda$ is a left artinian ring and for each indecomposable left $\Lambda$-module $M$, ${\rm Hom}_{\Lambda}(M,-) \simeq \varphi(-)$ for some positive-primitive formula $\varphi(x)$ over $\Lambda$.
\item[$(e)$] $\Lambda$ is a left artinian ring and every indecomposable projective object in  ${\rm Mod}((\Lambda$-mod$)^{\rm op})$ has finitely generated essential multiplicity-free socle.
\end{itemize}
\end{The}
\begin{proof}
$(a) \Rightarrow (b)$. Let $\Lambda$ be a left K\"othe ring. By \cite[Corollary 3.2]{za}, $\Lambda$ is of finite representation type. Let $\lbrace X_1, \cdots, X_m \rbrace$ be a complete set of representative of the isomorphic classes of finitely generated indecomposable left $\Lambda$-module. $T={\rm End}_{\Lambda}(X)$ is an Auslander ring of $\Lambda$, where $X=\bigoplus_{i=1}^mX_i$. We show that $T$ is a generalized right QF-2 ring.    Let $W$ be a minimal cogenerator left $\Lambda$-module and $\Delta={\rm End}_{\Lambda}(W)$. Since $\Lambda$ is of finite representation type, $W$ is a finitely generated left $\Lambda$-module. According to Remark \ref{r2}, ${\rm Hom}_{\Lambda}(-,W): \Lambda$-mod$\rightarrow$ mod-$\Delta$ is a duality and $\Delta$ is a right artinian ring. It is easily seen that $\lbrace {\rm Hom}_{\Lambda}(X_1,W), \cdots, {\rm Hom}_{\Lambda}(X_m,W) \rbrace$ is a complete set of representative of the isomorphic classes of finitely generated indecomposable right $\Delta$-modules. Then $\Delta$ is a ring of finite representation type and $\Theta={\rm End}_{\Delta}(\bigoplus_{i=1}^m{\rm Hom}_{\Lambda}(X_i,W))$ is the Auslander ring of $\Delta$. We conclude that the Auslander ring $T$ of $\Lambda$ is isomorphic to the Auslander ring $\Theta$ of $\Delta$ because ${\rm Hom}_{\Lambda}(X,W) \cong \bigoplus_{i=1}^m{\rm Hom}_{\Lambda}(X_i,W)$ as right $\Delta$-modules. Since the property "artinian generalized right QF-2" is Morita invariant, it is enough to show that $\Theta$ is a generalized right QF-2 ring. According to Proposition \ref{r1} it is enough to show that each ${\rm Hom}_{\Lambda}(X_i,W)$ has multiplicity-free socle. Let $1 \leq l \leq m$. top$(X_l)=\bigoplus_{i=1}^t S_i$, where $t \in {\Bbb{N}}$ and each $S_i$ is a simple left $\Lambda$-module. By \cite[Corollary 3.3]{za}, $X_l$ has multiplicity-free top and so $S_i \ncong S_j$ when $i \neq j$. It follows that ${\rm Hom}_{\Lambda}(S_i, W) \ncong {\rm Hom}_{\Lambda}(S_j, W)$ when $i \neq j$. On the other hand, by Remark \ref{r2}, soc$({\rm Hom}_{\Lambda}(X_l,W)) \cong {\rm Hom}_{\Lambda}({\rm top}(X_l),W)$ as $\Delta$-modules and hence  soc$({\rm Hom}_{\Lambda}(X_l,W)) \cong \bigoplus_{i=1}^t {\rm Hom}_{\Lambda}(S_i, W)$ as $\Delta$-modules. Therefore ${\rm Hom}_{\Lambda}(X_l,W)$ has multiplicity-free socle and the result follows.\\
$(b) \Rightarrow (c)$ is clear.\\
$(c) \Rightarrow (d)$. First we show that $\Lambda$ is a ring of finite representation type. Since $\Lambda$ is left artinian, there are only finitely many non-isomorphic simple left $\Lambda$-modules. Let $\lbrace S_1, \cdots, S_n \rbrace$ be a complete set of representative of the isomorphic classes of simple left $\Lambda$-modules. Let $\lbrace V_{\alpha}~|~ \alpha \in I\rbrace$ be a complete set of representative of the isomorphic classes of  finitely generated indecomposable left $\Lambda$-modules and $T=\widehat{{\rm End}}_{\Lambda}(V)$, where $V=\bigoplus_{\alpha \in I}V_{\alpha}$. Let $\alpha \in I$. Since $V_{\alpha}$ is a finitely generated left $\Lambda$-module, there exists a non-zero $\Lambda$-module epimorphism $h: V_{\alpha} \rightarrow S_i$ for some $1 \leq i \leq n$. We know that $V$ is a generator in $\Lambda$-Mod. Hence by \cite[Proposition 51.5(1)]{wi}, $\widehat{{\rm Hom}}_{\Lambda}(V,h): \widehat{{\rm Hom}}_{\Lambda}(V,V_{\alpha}) \rightarrow \widehat{{\rm Hom}}_{\Lambda}(V,S_i)$ is non-zero. Consequently there is a non-zero $T$-module epimorphism $\varphi:  Te_{\alpha} \rightarrow M$ where $M$ is a $T$-submodule of $\widehat{{\rm Hom}}_{\Lambda}(V,S_i)$. Since by the proof of Lemma \ref{l1}, ${\rm rad}(Te_{\alpha})$ is a maximal superfluous submodule of $Te_{\alpha}$,  we can see that $\varphi({\rm rad}(Te_{\alpha}))={\rm rad}(M)$.  On the other hand, since $T$ is a left locally noetherian ring, $\widehat{{\rm Hom}}_{\Lambda}(V,S_i)$ is a noetherian left $T$-module. It follows that $M$ is a finitely generated left $T$-module and so it has a maximal submodule. This means that ${\rm rad}(M)$ is a proper submodule of $M$. Therefore the $T$-module epimorphism  $\overline{\varphi}: Te_{\alpha}/{\rm rad}(Te_{\alpha}) \rightarrow M/{\rm rad}(M)$ defined by $x + {\rm rad}(Te_{\alpha}) \mapsto (x)\varphi + {\rm rad}(M)$ is  non-zero. Since ${\rm rad}(Te_{\alpha})$ is a maximal submodule of $Te_{\alpha}$,  $Te_{\alpha}/{\rm rad}(Te_{\alpha}) \cong M/{\rm rad}(M)$ as $T$-modules. Since $T$ is a left locally finite ring, $\widehat{{\rm Hom}}_{\Lambda}(V,S_i)$ has finite length. This implies that $M/{\rm rad}(M)$ is a composition factor of $\widehat{{\rm Hom}}_{\Lambda}(V,S_i)$. By Lemma \ref{l1}, $\lbrace Te_{\alpha}/{\rm rad}(Te_{\alpha})~|~\alpha \in I \rbrace$ is a complete set of representative of the isomorphic classes of simple unitary left $T$-modules. Therefore this set is finite. Consequently $\Lambda$ is a ring of finite representation type. Next we show that every finitely generated indecomposable left $\Lambda$-module has multiplicity-free top. Let $W$ be the minimal cogenerator in $\Lambda$-Mod. $W$ is finitely generated and by Remark \ref{r2}, the functor ${\rm Hom}_{\Lambda}(-,W): \Lambda$-mod$\rightarrow$ mod-$\Delta$ is a duality with the inverse duality ${\rm Hom}_{\Delta}(-,W):$ mod-$\Delta \rightarrow \Lambda$-mod, where $\Delta={\rm End}_{\Lambda}(W)$.  Moreover for each finitely generated left $\Lambda$-module $X$, we have soc$({\rm Hom}_{\Lambda}(X,W)) \cong {\rm Hom}_{\Lambda}({\rm top}(X),W)$ as $\Delta$-modules. Let $X$ be a finitely generated indecomposable left $\Lambda$-module. Then ${\rm top}(X)=\bigoplus_{j=1}^tS'_j$, where each $S'_j$ is a simple left $\Lambda$-module. It follows that soc$({\rm Hom}_{\Lambda}(X,W)) \cong \bigoplus_{j=1}^t{\rm Hom}_{\Lambda}(S'_j,W)$ as $\Delta$-modules. We observe that each ${\rm Hom}_{\Lambda}(S'_j,W)$ is a simple right $\Delta$-module. Since $T$ is generalized right QF-2, ${\rm Hom}_{\Lambda}(S'_j,W) \ncong {\rm Hom}_{\Lambda}(S'_k,W)$ when $j \neq k$. We conclude that $X$ has multiplicity-free top. Then by \cite[Corollary 3.2]{za}, every indecomposable left $\Lambda$-module is cyclic. Let $M$ be an indecomposable left $\Lambda$-module. Then there exists a $\Lambda$-module epimorphism $f: {_\Lambda\Lambda} \rightarrow  M$. It follows that there is a monomorphism ${\rm Hom}_{\Lambda}(M,-) \rightarrow {\rm Hom}_{\Lambda}({_\Lambda\Lambda},-)$ in Mod$((\Lambda$-mod$)^{\rm op})$. Therefore ${\rm Hom}_{\Lambda}(M,-) \simeq \varphi(-)$ for some positive-primitive formula $\varphi(x)$ over $\Lambda$. \\
$(d) \Rightarrow (a)$. Let $M$ be a finitely generated indecomposable left $\Lambda$-module and assume that there exists a monomorphism $\eta: {\rm Hom}_{\Lambda}(M,-) \rightarrow {\rm Hom}_{\Lambda}({_{\Lambda}\Lambda},-)$ in  Mod$((\Lambda$-mod$)^{\rm op})$. Yoneda's Lemma implies that $\eta={\rm Hom}_{\Lambda}(f,-)$ for some $\Lambda$-module homomorphism $f: {_{\Lambda}\Lambda} \rightarrow M$. It is easy to see that $f$ is an epimorphism. This yields that there is a finite upper bound for the lengths of finitely generated indecomposable left $\Lambda$-modules and hence by \cite[Proposition 54.3]{wi}, $\Lambda$ is of finite representation type. Therefore by \cite[Corollary 4.8]{ausla2}, $\Lambda$ is a left K\"othe ring and the result follows.\\
$(c) \Rightarrow (e)$. By using of the proof of $(c) \Rightarrow (d)$, we can see that $\Lambda$ is of finite representation type and hence $T$ is the Auslander ring of $\Lambda$. Also by \cite[Corollary 4.8]{ausla2}, every left $\Lambda$-module is a direct sum of finitely generated indecomposable left $\Lambda$-modules. By Proposition \ref{r3}, there exists an additive covariant equivalence $H:$ Mod-$T \rightarrow$ Mod$R$. It is easy to see that $H$ preserves essential monomorphisms and simple modules. By Remark \ref{r22}, the functor $\mathbf{F} \circ H :$ Mod-$T \rightarrow {\rm Mod}((\Lambda$-mod$)^{\rm op})$ is an additive equivalence of categories. Let $G$ be an indecomposable projective object in Mod$((\Lambda$-mod$)^{\rm op})$. By Remark \ref{r22}, there is an indecomposable projective unitary right $R$-module $X$ such that $\mathbf{F}(X) \cong G$.  Since by the proof of \cite[Theorem 3.10]{zn}, the functor $H$ reflects indecomposable projective objects, $H(Q) \cong X$ for some indecomposable projective right $T$-module $Q$. Since $T$ is an artinian ring, $Q$ is finitely generated and by \cite[Propositions 21.3 and 31.4]{wi}, soc$(Q)$ is finitely generated and essential in $Q$. Then soc$(Q)=\bigoplus_{i=1}^nS_i$, where $n \in {\Bbb{N}}$ and each $S_i$ is a simple right $T$-module. Since $T$ is a generalized right QF-2 ring, $S_i \ncong S_j$ when $i \neq j$. On the other hand  there is an essential monomorphism $h: H({\rm soc}(Q)) \rightarrow H(Q)$. We know that $H({\rm soc}(Q)) \cong \bigoplus_{i=1}^nH(S_i)$ as $R$-modules. Therefore ${\rm soc}(H(Q))=\bigoplus_{i=1}^nT_i$ is an essential submodule of $H(Q)$, where each $T_i \cong H(S_i)$ as $R$-modules. Moreover $T_i \ncong T_j$ when $i \neq j$. Now we have an essential monomorphism $\eta: \mathbf{F}({\rm soc}(H(Q))) \rightarrow \mathbf{F}(H(Q))$ in Mod$((\Lambda$-mod$)^{\rm op})$. Since $\mathbf{F}(H(Q)) \cong G$, there is an essential monomorphism $\tau: \mathbf{F}({\rm soc}(H(Q))) \rightarrow G$ in Mod$((\Lambda$-mod$)^{\rm op})$. So ${\rm Im}(\tau)$ is an essential subfunctor of $G$ which is isomorphism to $\mathbf{F}({\rm soc}(H(Q)))$. It is easy to see that $\mathbf{F}({\rm soc}(H(Q))) \cong \bigoplus_{i=1}^n \mathbf{F}(T_i)$ and also $\mathbf{F}(T_i) \ncong \mathbf{F}(T_j)$ when $i \neq j$. Moreover each $\mathbf{F}(T_i)$ is a simple object in  Mod$((\Lambda$-mod$)^{\rm op})$. Therefore $G$ has finitely generated essential multiplicity-free socle and the result follows.\\
$(e) \Rightarrow (a)$. Let $R$ be the left functor ring of $\Lambda$. Then every indecomposable projective unitary right $R$-module has finitely generated essential multiplicity-free socle. By Proposition \ref{p6}, $\Lambda$ is a left Morita ring. Choose $W$ the minimal cogenerator left $\Lambda$-module. By \cite[Proposition 47.15(2)]{wi}, $W$ is a finitely generated left $\Lambda$-module and $\Delta={\rm End}_{\Lambda}(W)$ is a right artinian ring. We show that every finitely generated indecomposable right $\Delta$-module has multiplicity-free socle. Let $N$ be a finitely generated indecomposable right $\Delta$-module. Since $\Delta$ is right artinian, by \cite[Propositions 21.3 and 31.4]{wi}, soc$(N)$ is finitely generated and essential in $N$. It follows that soc$(N)=\bigoplus_{i=1}^sT_i$, where $s \in {\Bbb{N}}$ and each $T_i$ is a simple right $\Delta$-module. Also $E(N) \cong \bigoplus_{i=1}^sE(T_i)$ as $\Delta$-modules. We show that $T_i \ncong T_j$ when $i \neq j$. Let $\lbrace U_{\alpha}~|~\alpha \in I \rbrace$ be a complete set of representative of the isomorphic classes of  finitely generated left $\Lambda$-modules such that $R=\widehat{{\rm End}}_{\Lambda}(U)$, where $U=\bigoplus_{\alpha \in I}U_{\alpha}$. By Remark \ref{r2}, the functor ${\rm Hom}_{\Lambda}(-,W): \Lambda$-mod $\rightarrow$ mod-$\Delta$ is a duality and hence $\lbrace {\rm Hom}_{\Lambda}(U_{\alpha},W)~|~ \alpha \in I \rbrace$ is a complete set of representative of the isomorphic classes of finitely generated right $\Delta$-modules. Then $T=\widehat{{\rm End}}_{\Delta}(V)$ is a right functor ring of $\Delta$, where $V=\bigoplus_{\alpha \in I}{\rm Hom}_{\Lambda}(U_{\alpha},W)$. Since $V$ is a generator in Mod-$\Delta$ we can see that $\widehat{{\rm Hom}}_{\Delta}(V,E(N))$ is injective in Mod$T$.  Also there is an essential monomorphism $\widehat{{\rm Hom}}_{\Delta}(V,N) \rightarrow \widehat{{\rm Hom}}_{\Delta}(V,E(N))$ in Mod$T$. Thus $E(\widehat{{\rm Hom}}_{\Delta}(V,N)) \cong \bigoplus_{i=1}^s \widehat{{\rm Hom}}_{\Delta}(V,E(T_i))$ as right $T$-modules. Since by Remark \ref{r2}, $W$ is an injective cogenerator right $\Delta$-module, each $E(T_i)$ is isomorphic to a direct summand of $W$. It follows that each $E(T_i)$ is finitely generated right $\Delta$-module. Since $\widehat{{\rm Hom}}_{\Delta}(V,-): {\rm Add}(V) \rightarrow {\rm Proj}(T^{\rm op})$ is an equivalence of categories, $\widehat{{\rm Hom}}_{\Delta}(V,N)$ and each $\widehat{{\rm Hom}}_{\Delta}(V,E(T_i))$ are projective unitary right $T$-modules. Moreover ${\rm End}_T(\widehat{{\rm Hom}}_{\Delta}(V,E(T_i))) \cong {\rm End}_{\Delta}(E(T_i))$ as rings. By \cite[Proposition 19.9]{wi},  ${\rm End}_T(\widehat{{\rm Hom}}_{\Delta}(V,E(T_i)))$ is a local ring.  On the other hand, by Remark \ref{r2}, $\Lambda$ is left Morita to $\Delta$. So we deduce from \cite[Lemma 3.1]{f} that every  indecomposable projective unitary right $T$-module has finitely generated essential multiplicity-free socle. This yields that ${\rm soc}(\widehat{{\rm Hom}}_{\Delta}(V,N))=\bigoplus_{i=1}^m{T'}_i$ with ${T'}_i \ncong {T'}_j$ when $i \neq j$, where $m \in {\Bbb{N}}$ and each ${T'}_i$ is a simple unitary right $T$-module. In addition $E(\widehat{{\rm Hom}}_{\Delta}(V,N)) \cong \bigoplus_{i=1}^m E({T'}_i)$ as $T$-modules. Therefore $\bigoplus_{i=1}^s \widehat{{\rm Hom}}_{\Delta}(V,E(T_i)) \cong \bigoplus_{i=1}^mE({T'}_i)$. Since ${\rm End}_T(E({T'}_i))$ is a local ring,  $m=s$ and for each $i$, $E({T'}_i) \cong \widehat{{\rm Hom}}_{\Delta}(V,E(T_i))$ as $T$-modules.  If $T_i \cong T_j$, then $E(T_i) \cong E(T_j)$. Thus $\widehat{{\rm Hom}}_{\Delta}(V,E(T_i)) \cong \widehat{{\rm Hom}}_{\Delta}(V,E(T_j))$. It follows that $E({T'}_i) \cong E({T'}_j)$ and hence $i=j$. Consequently every finitely generated indecomposable right $\Delta$-module has multiplicity-free socle. Let $M$ be a finitely generated indecomposable left $\Lambda$-module. Since $\Lambda$ is left artinian, top$(M)=\bigoplus_{i=1}^t S_i$, where $t \in {\Bbb{N}}$ and each $S_i$ is a simple left $\Lambda$-module. By Remark \ref{r2}, ${\rm soc}({\rm Hom}_{\Lambda}(M,W)) \cong \bigoplus_{i=1}^t{\rm Hom}_{\Lambda}(S_i,W)$ as $\Delta$-modules and ${\rm Hom}_{\Lambda}(M,W)$ is a finitely generated indecomposable right $\Delta$-module. Hence  ${\rm Hom}_{\Lambda}(S_i,W) \ncong {\rm Hom}_{\Lambda}(S_j,W)$ when $i \neq j$. This implies that $S_i \ncong S_j$ when $i \neq j$. Therefore every finitely generated indecomposable left $\Lambda$-module has multiplicity-free top. Consequently by  \cite[Corollary 3.2]{za}, $\Lambda$ is a left K\"othe ring and the result follows.
\end{proof}

As a consequence, we obtain the Auslander correspondence for left Kawada rings.

\begin{The}\label{ka}
There exists a bijection between the Morita equivalence classes of left Kawada rings and the Morita equivalence classes of Auslander generalized right QF-2 rings.
\end{The}
\begin{proof}
Let $\mathfrak{A}$ be the set of Morita equivalence classes of rings of finite representation type and $\mathfrak{B}$ be the set of Morita equivalence classes of Auslander rings. By \cite[Corollary 4.7]{ausla2},  the mapping $\varphi: \mathfrak{A} \rightarrow \mathfrak{B}$ via $\Lambda \mapsto {\rm End}_{\Lambda}(U)$ is a bijection, where $U$ is a direct sum of all modules in a complete set of representative of the isomorphic classes of  finitely generated indecomposable left $\Lambda$-modules. Let $\Lambda$ be a left Kawada ring. Since $\Lambda$ is a semiperfect ring, $\Lambda$ is Morita equivalent to $\Lambda_0$ for some basic ring $\Lambda_0$. $\Lambda_0$ is a left K\"othe ring and so by Theorem \ref{r33}, $\varphi(\Lambda_0)$ is a generalized right QF-2 ring. Since $\varphi(\Lambda)$ is Morita equivalent to $\varphi(\Lambda_0)$, we can see that the Auslander ring $\varphi(\Lambda)$ is a generalized right QF-2 ring. Let $R$ be an Auslander generalized right QF-2 ring. Then there exists a representation-finite ring $\Delta$ such that $\varphi(\Delta)$ is Morita equivalent to $R$. Since $\Delta$ is semiperfect, $\Delta$ is Morita equivalent to a basic ring $\Delta_0$. It follows that $\varphi(\Delta_0)$ is Morita equivalent to $\varphi(\Delta)$. This leads that $R$ is Morita equivalent to $\varphi(\Delta_0)$. Consequently $\varphi(\Delta_0)$ is a generalized right QF-2 ring. So we deduce from Theorem \ref{r33} that $\Delta_0$ is a left K\"othe ring. Hence by \cite[Corollary 4.2]{za}, $\Delta$ is a left Kawada ring. Therefore the mapping $\varphi$ induces a bijection between the Morita equivalence classes of left Kawada rings and the Morita equivalence classes of Auslander generalized right QF-2 rings.
\end{proof}

\section*{acknowledgements}
The research of the first author was in part supported by a grant from IPM. Also, the research of the second author was in part supported by a grant from IPM (No. 1400170417).

redpajama_set_name: RedPajamaArXiv