This result known as the Gershgorin circle theorem has been used as a basis for extension. In 1964 Brenner reported on "Theorems of Gersgorin Type". In 1967 at University of Wisconsin—Madison, working in the Mathematics Research Center, he produced a technical report "New root-location theorems for partitioned matrices". In 1968 Brenner, following Alston Householder, published "Gersgorin theorems by Householder’s proof". In 1970 he published the survey article (21 references) "Gersgorin theorems, regularity theorems, and bounds for determinants of partitioned matrices". The article was extended with "Some determinantal identities". In 1971 Brenner extended his geometry of the spectrum of a square complex matrix deeper into abstract algebra with his paper "Regularity theorems and Gersgorin theorems for matrices over rings with valuation". He writes, "Theorems can be extended to non-commutative domains, in particular to quaternion matrices. Secondly, the ring of polynomials has a valuation ... a different type of regularity ..." Joel Lee Brenner was a member of the American Mathematical Society from 1936. Beasley relates that he  In 1981 Brenner and Roger Lyndon collaborated to polish an idea due to H. W. Kuhn for proving the fundamental theorem of algebra. In the solution by Eric S. Rosenthal to a problem in the "American Mathematical Monthly" posted by Harry D. Ruderman, Kuhn's work from 1974 was cited. A query was made and prompted an article by Brenner and Lyndon. The version of the fundamental theorem stated was as follows:  Brenner ultimately acquired 35 coauthors in his publications. Given an ordered set Ω with "n" elements, the even permutations on it determine the alternating group An. In 1960 Brenner proposed the following research problem in group theory: For which An does there exist an element "a"n such that every element "g" is similar to a commutator of "a"n? Brenner states that the property is true for 4 < "n" < 10; in symbols it may be expressed The alternating groups are simple groups, and in 1971 Brenner began a series of articles titled "Covering theorems for finite simple groups". He was interested in the cycle type of cyclic permutations, and when An ⊂ "C C", where "C" is a conjugacy class of a certain type.