Muslim scholar Qadi 'Iyad comments on the meaning of the Muslim greeting in these words:
There are three possibilities contained by the expression, As-Salamu
'alaykum (Peace be upon you):
1) It means safety (salama) for you and with you - taking the word
salam (peace) as a verbal noun.
2) As-Salam means your protection, being guarded and your
preservation - in this case it is As-Salam as the name of Allah.
3) As-Salam in the meaning of musalama (reconciliation) and
submission to him in the words of Allah, "No, by your Lord, they will not
believe until they ask you to judge between them regarding what they
disagree about. Then they will not find any impediment in themselves about
your decision and will submit to you completely." (4:65)
(Muhammad Messenger of Allah (Ash-Shifa of Qadi 'Iyad),
Qadi 'Iyad Musa al-Yahsubi, translated by Aisha Abdarrahman Bewley
[Madinah Press, Inverness, Scotland, U.K. 1991; third reprint, paperback],
pp. 250-251)