Wa `alaykum As-Salamu waRahmatullahi wa Barakatuh.
In the Name of Allah, Most Gracious, Most Merciful.
All praise and thanks are due to Allah, and peace and blessings be upon His Messenger.
In this fatwa:
It is prescribed for the one who enters the mosque – whether he is the imam or one of the worshippers – to greet the people in the mosque with salam.
In this regard, Sheikh `Abdul-`Aziz ibn Baz, the late Mufti of Saudi Arabia, states:
He should greet the people in the mosque with salam. If there is someone in the mosque, he should greet him with salam before he starts to pray. He should say salam, then start his prayer, whether it is a prayer to greet the mosque (tahiyyat al-masjid) or regular Sunnah prayer.
If the people in the mosque are busy, he should still greet them with salam, even if they are reading Qur’an. The one who is reading can pause, say “Wa ‘alaykum as-salam,” then resume his reading. (Fatawa Nur ‘ala ad-Darb, 11/344)
Allah Almighty knows best.