Gemara

(R. Yehoshua ben Levi): When Moshe ascended to receive the Torah, he found Hash-m tying crowns to the letters.

Hash-m: People don't give Shalom in your city?! (Why didn't you greet Me?)

Moshe: Does a slave give Shalom to his master?!

Hash-m: You should help Me (give Me a Berachah).

Moshe: "V'Atah Yigdal Na Ko'ach Hash-m ka'Asher Dibarta."

Berachos 3a (Beraisa - R. Yosi): Once, when I was traveling, I entered a ruin in Yerushalayim to pray. Eliyahu waited for me at the door.

Eliyahu: Shalom Alecha Rebbi!

R. Yosi: Shalom Alecha Rebbi u'Mori.

14a (Beraisa): If one was between the Perakim of Shema, and then came his Rebbi or someone more important than himself, he greets him.

27b (Beraisa - R. Eliezer): One who gives or returns Shalom to his Rebbi causes the Shechinah to depart from Yisrael.

Nazir 20b (Mishnah): If Reuven said 'I am a Nazir'; another (heard him and) said 'and I'; (and another said) 'and I', they are all Nezirim.

(Reish Lakish): This is when each said 'and I' Toch Kedei Dibur.

This is the time needed for a Talmid to greet his Rebbi.

Objection (R. Yehudah Nesi'ah): This does not leave time for a Talmid to greet his Rebbi, and then say 'and I'!

(Beraisa): Kedei Dibur is the time for a Talmid to greet his Rebbi.

Sanhedrin 98a: R. Yehoshua ben Levi (to Mashi'ach): Shalom Alecha Rebbi u'Mori!

Bava Kama 73b: There are two measures of Toch Kedei Dibur: a Talmid greeting his Rebbi, and a Rebbi greeting his Talmid;

A Rebbi greets his Talmid 'Shalom Alecha' (two words). R. Yosi holds that within this time is a continuation;

A Talmid greets his Rebbi 'Shalom Alecha Rebbi u'Mori' (four words). R. Yosi holds that within this time (but after the time for two words) is not a continuation.

Yerushalmi Berachos 13a (Beraisa): (Between the Perakim of Shema) one may give Shalom to his Rebbi or someone greater in Torah than himself.

Inference: (When one is not amidst Shema) one must give Shalom to someone greater in Torah than himself.

We learn this also from the following.

(Beraisa): If one tore [his clothing over the death of a relative], and the relative revived [and died again] afterwards, he must tear again.

"Afterwards" is more than Toch Kedei Dibur later.

(R. Simon citing R. Yehoshua ben Levi): This is the time to greet a colleague.

(Aba bar bar Chanah citing R. Yochanan): This is the time to give Shalom between a Rav and his Talmid - Shalom Alecha Rebbi.

R. Yochanan was walking, supporting himself on R. Yakov bar Idi. R. Elazar saw him and hid from him.

R. Yochanan: This Babylonian did two things. He did not give Shalom to me, and he did not recite my teaching in my name!

R. Yakov bar Idi: This is their custom;

Ze'ira did not give Shalom to Rabah. They fulfilled "Ra'uni Ne'arim v'Nechba'u".

Rishonim

Rif (Berachos 18b): One who gives Shalom to his Rebbi causes the Shechinah to depart from Yisrael.

R. Yonah (DH veha'Nosen): This connotes that the Talmid greets the Rebbi first. This is improper. He should unable to speak up first due to his great awe, like Moshe said "does a slave give Shalom to his master?!" In many places, the Gemara discusses She'elas Shalom of a Talmid to his Rebbi. It really refers to returning Shalom.

Note: Our text says that also returning Shalom causes the Shechinah to depart.

Rambam (Hilchos Talmud Torah 5:5): One does not give Shalom to his Rebbi or answer like we do to friends. Rather, he bows and says with awe and honor 'Shalom Alecha Rebbi.' If his Rebbi gave Shalom to him, he answers 'Shalom Alecha Rebbi v'Mori.'

Rosh (Berachos 4:5): Rashi says that the Isur is to give Shalom to one's Rebbi like he does to others, without saying 'Rebbi'. The Yerushalmi says that a smaller Chacham does not greet a bigger Chacham at all, to fulfill "Ra'uni Ne'arim v'Nechba'u". However, it seems that the Bavli argues, for it says that Toch Kedai Dibur is the time to give Shalom to one's Rebbi (Bava Kama 73b).

Rebuttal (second answer in Tosfos Berachos 27b DH veha'Nosen): Only a Talmid Chaver may give Shalom to his Rebbi. (A Talmid Chaver is a Chacham who can reason by himself, but he needs to hear teachings from his Rebbi. Rashi says that he is a Chacham like the Rebbi, just he learned from him one or more things.)

Defense (of Rosh - Gra YD 242:39): In the Yerushalmi, R. Yochanan complained that Benei Bavel do not give Shalom to him. R. Yakov bar Idi explained that they hold that a smaller Chacham does not greet a bigger Chacham, to fulfill "Ra'uni Ne'arim v'Nechba'u". The Rosh learns from the Shi'ur of Toch Kedai Dibur that the Bavli argues. R. Yonah says that the Beraisa forbids giving Shalom, but permits returning Shalom. Really, the Yerushalmi proves oppositely. R. Yochanan was upset that they did not greet him. It is difficult to say that he did not know the Beraisa. Also, why didn't R. Yakov bar Idi answer that they hold like the Beraisa! Rather, only they (Bnei Bavel) do not greet, due to their custom.

Question: In Shabbos, it says that a slave does not give Shalom to his master!

Answer (Tosfos 73b DH Kedei): A slave is different than a Talmid. A slave fears his master, therefore he should never give Shalom.

Rashi (27b DH Shalom): The Isur is to give Shalom to his Rebbi like to others "Shalom Alecha", without saying "Shalom Alecha Rebbi."

Poskim

Shulchan Aruch (YD 242:16): One does not give Shalom to his Rebbi like he does to others. Rather, he bows and says with awe and honor 'Shalom Alecha Rebbi.' If his Rebbi gave Shalom to him, he answers 'Shalom Alecha Mori v'Rebbi.'

Source (Gra 38): We learn from Sanhedrin 98a. This needs investigation.

Note: R. Yehoshua ben Levi greeted Moshi'ach with 'Shalom Alecha Rebbi v'Mori.' The Gra's question assumes that one need not show more honor to Mashi'ach when greeting him than to one's Rebbi.)

Taz (5) and Shach (25): If his Rebbi gave Shalom to him, he answers Shalom Alecha Rebbi u'Mori. The Rambam and Tur say so. This is because it is normal to add when returning Shalom (148:10). Also the Mechaber says so in old editions of the Shulchan Aruch.

R. Akiva Eiger: I say that the Rambam learned from Berachos 3a.

Rema: This is the custom. Some say that a Talmid does not give Shalom at all - "Ra'uni Ne'arim v'Nechba'u."

Panim Me'iros (introduction to Chelek 1, DH Temu'ah): The Rema is astounding. This is from the Rosh, who learned from the Yerushalmi. Also R. Yonah said that a Talmid should not give Shalom first, due to his great awe of his Rebbi. The Yerushalmi connotes oppositely! It teaches that one must greet his Rebbi. This is clear from R. Yochanan. Even R. Elazar hid from R. Yochanan so he would not see him. According to R. Yonah, he did not need to hide, for in any case one may not give Shalom to his Rebbi! Rather, R. Elazar did not give Shalom because he hid. If one passes in front of his Rebbi and does not give Shalom with fear and honor "Shalom Alecha Rebbi u'Mori", this is crude. A Stam Mishnah (Berachos 14a) permits interrupting Shema to greet his Rebbi. If one may not give Shalom to his Rebbi, this (not giving Shalom) shows fear. Why did Chachamim permit interrupting in the middle of a Perek due to fear of his Rebbi?! Rather, it is like I wrote. The Yerushalmi's proof from She'elas Shalom of a Talmid to his Rebbi is difficult. Perhaps it is Reshus [to give Shalom to his Rebbi]. What is the source that it is a Chiyuv?! Tosfos (Bava Basra 129b DH v'Hilchesa) and the Ran (Nedarim 87a DH v'Hilchesa, citing R. Tam) say that Chachamim enacted that Toch Kedei Dibur is like [part of the same] Dibur, so that if a Talmid is buying something and his Rebbi passes, he will be able to greet him and not lose his purchase. This proves that one must greet his Rebbi. If not, Chachamim would not have enacted.

Birkei Yosef (19): It is also difficult for the Rosh, who says that the Yerushalmi holds that a Talmid does not give Shalom to his Rebbi at all, for the Yerushalmi learns from the Shi'ur of Toch Kedei Dibur that a Talmid greets his Rebbi! It explicitly says that the Talmid greets first, unlike R. Yonah. This requires resolution.

R. Akiva Eiger: The Rambam (Hilchos Edus 20:3) mentioned She'elas Shalom of a Talmid to his Rebbi, but did not give the Shi'ur. Perhaps he relied on what he wrote in Hilchos Talmud Torah, that it is Shalom Alecha Rebbi. In Bava Kama, we say that it is Shalom Alecha Rebbi u'Mori! Perhaps the Rambam had a different text. It is astounding that the Acharonim did not discuss this.

Yad Avraham: Berachos 62a proves that it suffices to say "Rebbi Ploni." (R. Akiva said 'R. Yehoshua', and Ben Azai said 'R. Akiva.')