What is the significance of the phrase "peace be upon him" (عليه السلام) that is used after mentioning prophets? Is it a salutation or a supplication, and what is the need for it given that the prophets are alive with their Lord and receiving sustenance?

"Peace" (As-Salam) is one of the names of Allah, meaning freedom from imperfections. Our saying "peace be upon him" (alayhi as-salam) is a supplication for the Prophet for protection from all harm and an increase in his elevation and honor.

As for the hadith, "Do not say 'Upon you be peace,' for 'Upon you be peace' is the greeting of the dead," it does not contradict this meaning. This prohibition does not mean that the prescribed greeting for the dead is "Upon you be peace." Rather, it is an informative statement about a prevalent custom among poets and others of greeting the dead with this phrase, and the Prophet, peace be upon him, disliked being greeted with the greeting of the dead.

The Sunnah for greeting the living and the dead is to say "Peace be upon you" (as-salamu alaykum); that is, by presenting the supplication before the name of the one being addressed.