Praise be to Allaah.

The tasleem or salaam at the end of the prayer is one of its pillars or essential parts, and one cannot end or exit from the prayer without saying it, because of the report narrated by Abu Dawood (61) and at-Tirmidhi (3) from ‘Ali (may Allah be pleased with him) who said: The Messenger of Allah (blessings and peace of Allah be upon him) said: “The key to prayer is purification, its start is the takbeer (saying Allahu akbar) and its exit is the tasleem (saying the salaam).”

Classed as saheeh by al-Albaani in Sunan Abi Dawood. 

This is the view of a number of scholars, including the Imams Maalik, ash-Shaafa‘i and Ahmad. 

An-Nawawi (may Allah have mercy on) said: 

Our view is that it is obligatory, one of the pillars (essential parts) of prayer without which it is not valid. This is the view of the majority of scholars among the Sahaabah, Taabi‘een and those who came after them. End quote. 

Al-Majmoo‘, 3/462 

When saying the salaam, the worshipper should intend to exit the prayer and say salaam upon the imam and those to his right and his left, and to the recording angels. 

An-Nawawi (may Allah have mercy on him) said: 

When saying the salaam, the imam should intend the first salaam as an exit from the prayer and a greeting to those who are on his right and upon the recording angels, and he should intend the second salaam as a greeting to those on his left and the recording angels. The one who is praying behind him should intend the first salaam as an exit from the prayer and as a greeting to the imam, the recording angels and the worshippers in his vicinity, in his row and behind him and in front of him, and he should intend the second salaam as a greeting to the worshippers in his vicinity, and if the imam is in front of him, he should intend his two salaams to be greetings to him too. 

The one who is praying alone should intend his first salaam as an exit from the prayer and a greeting to the recording angels, and the second salaam as a greeting to the recording angels. The basic principle in his case is that which was narrated by Samurah ibn Jundub (may Allah be pleased with him) who said: The Messenger of Allah (blessings and peace of Allah be upon him) instructed us to send salaams upon ourselves and to greet one another with salaam… 

If he intends to exit the prayer and does not intend anything else, that is permissible because saying salaam to those who are present is Sunnah.

End quote from al-Majmoo‘, 3/456 

Muslim (431) narrated that Jaabir ibn Samurah (may Allah be pleased with him) said: When we prayed with the Messenger of Allaah (blessings and peace of Allah be upon him) we used to say, Al-saalaamu ‘alaykum wa rahmat-Allaah, al-salaamu ‘alaykum wa rahmat-Allaah (Peace be upon you and the mercy of Allaah), and he gestured with his hand to either side. The Messenger of Allaah (blessings and peace of Allah be upon him) said: “Why do you gesture with your hands as if they were the tails of restive horses? Rather it is sufficient for one of you to put his hand on his thigh then say salaams to his brothers to his right and left.” 

An-Nawawi (may Allah have mercy) said: 

The words of the Prophet (blessings and peace of Allah be upon him), “then say salaams to his brothers to his right and left” refer to his fellow worshippers who are present on his right on his left. End quote. 

Ibn Qudaamah (may Allah have mercy on him) said:

He should intend when saying the salaam, to exit the prayer; if he also intends alongside that to respond to the two angels and to those who are behind him, if he is the imam, or to the imam and the other people with him, if he is a member of the congregation, there is nothing wrong with that. This was stated by Ahmad who said: He may say the salaam at the end of the prayer, intending thereby to respond to the imam. End quote. 

Al-Mughni, 1/326-327 

Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said: 

If it is asked: upon whom are salaams being sent? 

The answer is that they said: If there is a group of others with him, then the salaam is upon them; if there is no group with him, then the salaam is upon the angels who are on his right and his left, so he says: As-salaamu ‘alaykum wa rahmat-Allah. End quote. 

Ash-Sharh al-Mumti‘, 3/208 

To sum up: 

When saying the salaam at the end of the prayer, the worshipper intends three things: 

-         To exit the prayer

-         To send salaams upon the recording angels

-         To send salaams upon his fellow-worshippers. 

If he is praying alone, then he intends when saying the salaam to exit the prayer and send salaams upon the recording angels. 

And Allah knows best.