The ruling on a Muslim initiating the greeting of peace.

The Shaykh : So what is the ruling on initiating the greeting of peace? Is it like returning the greeting?
There is no doubt that returning the greeting, as the evidences have agreed upon, is stronger in the ruling of the Shari'ah than initiating the greeting. This is because returning the greeting is obligatory by this Qur'anic text, and because the hadith that came from the Prophet (peace and blessings be upon him) is authentic: The first thing that appears and comes to mind regarding this stinginess is his returning of the greeting that was offered to him by his Muslim brother. As for initiating the greeting, it is undoubtedly a recommended practice; this is according to the weakest of the opinions. And the stronger opinion is—there is no room left, my brother; so fix your head—the stronger opinion is that initiating the greeting is also obligatory, but within the limits of one's ability. Because the hadith says: If we were to imagine a man who leaves his house and every time he meets a Muslim he offers him the greeting, he would fall into hardship. This is not possible, and hardship is removed by the text of the Qur'an. Therefore, the best, according to what my research and knowledge have reached, is to say that initiating the greeting is also obligatory, but within the limits of what is easy and possible, and what is not accompanied by hardship.
Returning the greeting and initiating the greeting are a right of the Muslim upon his Muslim brother.