Is Salamun `Alaykum Correct?
Question

Many Muslims greet their brothers by saying “Salamun `alaykum”. Is it permissible for us to say that? If it is not correct, will the one who does that still be rewarded for saying Salam?.
Praise be to Allah.Is greeting with Salamun `Alaykum Correct?
There is nothing wrong with the person who is initiating the greeting saying, “Salamun `alaykum” or `Salamun `alayka.” Allah tells us that the greeting of the angels to the people of Paradise will be “Salamun `alaykum ” as He says (interpretation of the meaning):
“And angels shall enter unto them from every gate (saying): `Salamun `Alaykum (peace be upon you) for you persevered in patience! Excellent indeed is the final home!’” [Ar-Ra`d 13:23, 24]
“And those who kept their duty to their Lord (Al-Muttaqun – the pious) will be led to Paradise in groups till when they reach it, and its gates will be opened (before their arrival for their reception) and its keepers will say: Salamun `Alaykum (peace be upon you)! You have done well, so enter here to abide therein.” [Az-Zumar 39:73]
The Salam in this form is also mentioned in the verse where Allah says (interpretation of the meaning):
“Those whose lives the angels take while they are in a pious state (i.e. pure from all evil, and worshipping none but Allah Alone) saying (to them): Salamun `Alaykum (peace be on you) enter you Paradise, because of that (the good) which you used to do (in the world).” [An-Nahl 16:32]
“And when they hear Al-Laghw (dirty, false, evil or vain talk), they withdraw from it and say: `To us our deeds, and to you your deeds. Peace be to you (Salamun `Alaykum). We seek not (the way of) the ignorant.’” [Al-Qasas 28:55]
“When those who believe in Our Ayaat (proofs, evidences, verses, lessons, signs, revelations, etc.) come to you, say: Salamun `Alaykum (peace be on you); your Lord has written (prescribed) Mercy for Himself, so that if any of you does evil in ignorance, and thereafter repents and does righteous good deeds (by obeying Allah), then surely, He is Oft-Forgiving, Most Merciful.” [Al-An`am 6:54]
Ibn Hibban narrated (493) from Abu Hurayrah (may Allah be pleased with him) that a man passed by the Messenger of Allah (peace and blessings of Allah be upon him) when he was in a gathering and said, “Salamun `alaykum.” He (the Prophet (peace and blessings of Allah be upon him)) said, “Ten rewards.” Another man passed by and said, “Salamun `alaykum wa rahmatu-Allahi (Peace be upon you and the mercy of Allah).” He said, “Twenty rewards.” Then another man passed by and said, “Salamun `alaykum wa rahmatu-Allahi wa barakatuhu (Peace be upon you and the mercy of Allah and His blessings).” He said: “Thirty rewards.” A man got up and left the gathering and did not say Salam. The Prophet (peace and blessings of Allah be upon him) said, “How soon your companion has forgotten. When one of you comes to a gathering and says Salam, if he wants to sit down then let him do so. Then when he wants to leave, let him say Salam, for the one is not more important than the other.” (Classed as authentic by Al-Albani in Sahih At-Targhib wal-Tarhib, 2712)
This and other evidence indicates that there is nothing wrong with saying Salam to a person by saying “Salamun `alaykum.”  He will be rewarded for that, and he deserves a response to his Salam.
Which is better: As-Salamu `alaykum or Salamun `alaykum?
The scholars differed as to which version is better, As-Salamu `alaykum or Salamun `alaykum, or whether they are both the same.
Al-Mardawi (may Allah have mercy on him) said in Al-Insaf (2/563): “If a person greets a living person, then the correct view in our Madhhab is that he has the choice between using the definite form (with Al-) or the indefinite (without Al-). And he said: this was stated by more than one.”
Then he mentioned a report from Imam Ahmad (may Allah have mercy on him) which says that the definite form is better than the indefinite, and he quoted Ibn `Aqil (may Allah have mercy on him) as saying that the indefinite is better than the definite.
Al-Nawawi (may Allah have mercy on him) said in Al-Adhkar (p. 356-358):
“Note that it is preferable for the Muslim to say “As-Salamu `alaykum wa rahmatu-Allahi wa barakatuhu”, using the plural pronoun (-kum), even if he is greeting only one person. The respondent should say, “Wa `alaykum As-Salam wa rahmatu-Allahi wa barakatuhu.”
Our companions said: If the one who is initiating the greeting says, “As-Salamu `alaykum,” then he has given the greeting. If he said “As-Salamu `alayka” or “Salamun `alayka” this also counts as the greeting.
As for the response, the minimum is to say, `Wa `alayka As-Salam” or “Wa `alaykum As-Salam.” If the waw is omitted and one says, “`Alaykum As-Salam” that is sufficient and is a response.
If the person who is initiating the greeting says “Salamun `alaykum” or “As-Salamu `alaykum,” in both cases the respondent may say, “Salamun `alaykum” , or he may say, “As-Salamu `alaykum”. Allah says (interpretation of the meaning):
“When they came in to him and said: `Salam, (peace be upon you)!’ He answered: `Salam, (peace be upon you)’” [Adh-Dhariyaat 51:25]
Imam Abul Hasan Al-Wahidi (may Allah have mercy on him), one of our companions, said: You have the choice between using the definite form (As-Salam) or the indefinite (Salamun).
I (An-Nawawi) say: But the alif and lam are preferable (i.e., saying the word Salam with the definite article Al-).”
Is it permissible to initiate the greeting by saying `Alayk As-Salam or `Alaykum As-Salam? 
What is disliked is for the one who is initiating the greeting to say ‘`Alayka As-Salam’ or ‘`Alaykum As-Salam,’ because this is the greeting given to the dead as the Prophet (peace and blessings of Allah be upon him) said.
Abu Dawud (5209) and At-Tirmidhi (2722) narrated that Abu Jurayy Al-Hujaymi (may Allah be pleased with him) said: I came to the Prophet (peace and blessings of Allah be upon him) and said: “ `Alayka As-Salam (upon you be peace), O Messenger of Allah.” He said, “Do not say ` `Alayka As-Salam’, for `Alayka As-Salam’ is the greeting of the dead.” This Hadith was classed as authentic by Al-Albani in Sahih Abu Dawud.
What is meant by the Prophet’s words “`Alayka As-Salam’ is the greeting of the dead” is the fact that many poets and others greeted the dead in this manner. Moreover the Sunnah of the Prophet (peace and blessings of Allah be upon him) is to greet the dead in the same manner as he greeted the living, by saying “As-Salamu `alaykum.”
Ibn Al-Qayyim (may Allah have mercy on him) said, explaining that:
“His greeting, when he was initiating the greeting, was to say, “As-Salamu `alaykum wa rahmatu-Allah”. He did not like the initiator to say “`Alayka’l-Salam.” Abu’l-Jurayy Al-Hujaymi said: I came to the Prophet (peace and blessings of Allah be upon him) and said: “ `Alayka As-Salam (upon you be peace), O Messenger of Allah.” He said, “Do not say ‘`Alayka As-Salam’, for `Alayka As-Salam’ is the greeting of the dead.” (This Hadith was classed as authentic by Al-Albani in Sahih Abu Dawud)
This Hadith confused some people, who thought that it contradicted what was narrated from the Prophet (peace and blessings of Allah be upon him) about greeting the dead by saying, “As-Salamu `alaykum,” saying the word Salam first. They thought that his words, “`Alayka As-Salam’ is the greeting of the dead” was describing what is prescribed, but that is a mistake which led them to think that there was a contradiction. Rather what is meant by the words “`Alaykas-Salam’ is the greeting of the dead” is a statement of something that used to happen, not something that was prescribed, namely that the poets and others used to greet the dead in this manner, and he (peace and blessings of Allah be upon him) did not like the greeting commonly used for the dead to become widespread.” (Zad Al-Ma`ad, 2/383)
What is the best way to greet someone in Islam? 
The most complete form of greeting is to say “As-Salamu `alaykum wa rahmatu-Allahi wa barakatuhu (Peace be upon you and the mercy of Allah and His blessings)” or “Salamun `alaykum wa rahmatu-Allahi wa barakatuhu”, because of the Hadith of Ibn Hibban quoted above, and because of the Hadith narrated by Abu Dawud (5195) and At-Tirmidhi (2689) from `Imran ibn Husayn (may Allah be pleased with him) who said: A man came to the Prophet (peace and blessings of Allah be upon him) and said: “As-Salamu `alaykum,” and he returned the greeting, then he sat down. The Prophet (peace and blessings of Allah be upon him) said, “Ten.” Then another man came and said, “As-Salamu `alaykum wa rahmatu-Allah,” and he returned the greeting, then he sat down. He said, “Twenty.” Then another man came and said As-Salamu `alaykum wa rahmatu-Allahi wa barakatuhu,” and he returned the greeting, then he sat down. He said, “Thirty.” (Classed as authentic by Al-Albani in Sahih Abi Dawud)
As for adding the words “wa maghfiratuhu (and His forgiveness)” or “wa ridwanuhu (and His pleasure)”, these are not mentioned in any authentic report from our Prophet (peace and blessings of Allah be upon him), as was stated by Ibn Al-Qayyim in Zad Al-Ma`ad (2/381) and by Al-Albani in Da`if Abu Dawud (5196).
And Allah knows best.