Praise be to Allah the Lord of the Worlds. May His peace and blessings be upon our Prophet Mohammad and upon all his family and companions.
Generally speaking, As-Salaam-Alaikum is the standard greeting of Islam and it is recommended for a Muslim to say As-Salaam-Alaikum upon entering a mosque or any other place. The proof of this is that Narrated Abu Huraira: A man entered the mosque and started praying while Allah's Messenger (PBUH) was sitting somewhere in the mosque. Then (after finishing the prayer) the man came to the Prophet (PBUH) and greeted him. The Prophet (PBUH) said to him, "Go back and pray, for you have not prayed. The man went back, and having prayed, he came and greeted the Prophet. The Prophet (PBUH) after returning his greetings said, "Go back and pray, for you did not pray." On the third time the man said, "(O Allah's Messenger (PBUH)!) teach me (how to pray)." The Prophet said, "When you get up for the prayer, perform the ablution properly and then face the Qibla and say Takbir (Allahu Akbar), and then recite of what you know of the Qur'an, and then bow, and remain in this state till you feel at rest in bowing, and then raise your head and stand straight; and then prostrate till you feel at rest in prostration, and then sit up till you feel at rest while sitting; and then prostrate again till you feel at rest in prostration; and then get up and stand straight, and do all this in all your prayers." {Transmitted by Bukhari}.
Ibn Hajar Al-Askalani {May Allah have mercy on him} said: "If upon entering the mosque he found others already present, he should greet them before performing prayer."{Fateh Al-Bari, Vol.7:170}. An-Nawawi {May Allah have mercy on him} said: "Upon entering one`s house, someone else's or a mosque, it is recommended to say As-Salaam-Alaina Wa Ala Eibad Allah As-Saliheen {Peace be upon us and upon God’s upright servants (for when he says that it reaches every upright servant in heaven and earth) }, even if nobody was present." {Al-Majmou` Shareh Al-Mohathab, Vol.4:612}.
Moreover, returning the greeting is a collective obligation, so if one or a group of praying people greeted back, then other praying people are cleared from liability before Almighty Allah. In case the person entering the mosque feared to annoy the praying people or those reciting Quran, then he should greet in a low voice enabling those close to him to hear him and return the greeting. Al-Shafi`I {May Allah have mercy on him} said: "If upon entering the mosque during Friday sermon a man greeted those already sitting in the mosque, then this is disliked for him. However, some of those sitting in the mosque should return the greeting because returning the greeting is an obligation." {Al-Umm, 234}.
In conclusion, upon entering the mosque, it is permissible to greet those already sitting and they should return the greeting. However, if it was feared that this could annoy the prayer performers, then one or some of them should return the greeting. In addition, if he, upon entering the mosque, didn`t greet those already sitting for fear of causing inconvenience, then he isn`t considered sinful because, in the first place, greeting is an act of Sunnah {recommended}. And Allah the Almighty knows best.
