Our talk here shall be about Adhkar which a Muslim says when he finishes his Salah; and many Hadiths were reported in this regard.
Of which is the Hadith that was reported by Muslim in his Sahih on the authority of Thawban (may Allah be pleased with him) that the Messenger of Allah (peace be upon him) said: "When the Messenger of Allah (peace be upon him) finished his Salah, he begged forgiveness three times and said: "Allahumma Antas-Salam, wa Minkas-Salam, Tabarakta Dhal-Jalali wal Ikram." O Allah! You are the Peace, and peace comes from You; Blessed are You, O Possessor of Glory and Honor. Al Walid (one of the narrators of the Hadith) reported: I said to Awza`y: How is the seeking of forgiveness? He replied: You should say: "Astagfirullaha, Astagfirullaha." I beg forgiveness from Allah, I beg forgiveness from Allah." [Sahih Muslim, No. 591].
As for the Prophet's saying: "Allahumma Antas-Salam," As-Salam is one of the beautiful names of Allah which Allah commanded us to use in invocation, as He says: "And (all) the Most Beautiful Names belong to Allâh, so call on Him by them." [Surat Al A`raf: 180]. And the meaning is: He is glorified from every defect or shortage and He is glorified to have any attribute that contradicts His Attributes of perfection, and He is glorified to have a counter or a parallel.
As for Prophet's saying: "wa Minkas-Salam," i.e., safety of danger is sought from You alone and is not sought from anyone else.
As for the Prophet's saying: "Tabarakta Dhal-Jalali wal Ikram," Tabarakta means: You are exalted and glorified. "Dhal-Jalali wal Ikram," it means: You are the owner of glory and honor which are two great attributes to Allah that indicate the perfection of His Greatness, Pride, and Glory. It also indicates His Great Attributes and the multiplicity of His Favors which entail that peoples' hearts should be filled by love, glory, and reverence to Him.
The wisdom behind asking for forgiveness after Salah is that a person should show that he has not performed his Salah as should and did not reach perfection in worship, but he must have done it imperfectly and he has been negligent; so he has to seek forgiveness.
Then a Musalli (one who offers Salah) should engage himself in Tahlil (Saying: La Ilaha Illa Allah i.e., No god but Allah). It was reported on the authority of the freed slave of Al Mughirah ibn Shu`bah who said: Once Al Mughirah dictated to me in a letter addressed to Mu`awiyah that the Prophet (peace be upon him) used to say after every compulsory Salah, “La Ilaha Ilallah Wahdahu La Sharika Lahu, Lahul-Mulku Walahul-Hamdu, Wahuwa `Ala Kulli Sha'in Qadir. Allahumma La Mani`a Lima A`tayta, Wa La Mu`tiya Lima Mana`ta, Wa La Yanfa`u Dhal-jaddi Minkal-jadd. (None has the right to be worshipped but Allah and He has no partner in Lordship or in worship or in the Names and the Qualities, and for Him is the Kingdom and all the praises are for Him and He is omnipotent. O Allah! Nobody can hold back what you give and nobody can give what You hold back. Hard efforts by anyone for anything cannot benefit one against Your Will).” [Reported by Al Bukhari in his Sahih, No. 844, and Sahih Muslim, No. 593].
`Abdullah ibn Az-Zubayr (may Allah be pleased with him) narrated that he used to say after each: "La Ilaha Illa Allah Wahdahu La Sharika Lah, Lahul-Mulku Wa Lahul-Hamd, Wa Huwa `Ala Killi Shay'in Qadir. La Hawla wala Quwwata Illa Billah, La Ilaha Illa Allah, Wa La Na`buda Illa Iyyah. Lahun-Ni`mah Wa Lahul-Fadl Wa Lahuth-Thana'ul-Hasan. La Ilaha Illa Allahu Mukhlisina Lahud-Dina Wa Law Karihal-Kafirun." There is none worthy of worship (in truth) but Allah. He is alone. There is no partner with Him. Sovereignty belongs to Him and He is Potent over everything. There is no might or power except with Allah. there is none worthy of worship (in truth) but Allah and we do not worship but Him alone. To Him belong all bounties, to Him belongs all Grace, and to Him is worthy praise accorded. there is none worthy of worship (in truth) but Allah, to Whom we are sincere in devotion, even though the unbelievers should disapprove it." (The narrator said): He (the Prophet) uttered it at the end of every (obligatory) Salah. [Reported by Muslim, No. 594].
As for the Prophet's saying: "Wa La Yanfa`u Dhal-Jaddi Minkal-Jadd," i,e., the rich man shall not benefit from his wealth before You, but he only benefits from his faith and from the obedience of His Commands.
As for the Prophet's saying: "La Ilaha Illa Allahu Mukhlisina Lahud-Dina Wa Law Karihal-Kafirun," i.e., we keep adherent to this monotheism and sincerity even if polytheists hate this.
It is permissible for a Muslim to recite the reported glorifications which the Prophet (peace be upon him) used to say after Salah.