Rav and the post-Mishnaic world

Rav Abba said in the name of Rav Huna: We made ourselves in Bavel equal to the land of Israel in terms of divorces from the time that Rav came to Bavel

He (Rabbi Yochanan) asked of him (Isi Bar Hini): Who is the head of the academy in Bavel? He answered him: Abba Aricha (Rav's real name) He said to him: You only call him Abba Aricha? I remember when I would sit before Rebbe (Yehudah the Prince), I was seventeen rows behind Rav, seeing sparks of fire leaping from the mouth of Rebbe into the mouth of Rav, and from the mouth of Rav into the mouth of Rebbe, and I could not understand what they were saying. And you simply call him Abba Aricha??

When Rabbah Bar Chana went to Bavel, Rabbi Chiya said to Rebbe: My brother's son is going to Bavel. May he decide in matters of ritual law? He may. May he decide in matters of monetary law? He may. May he decide that first born animals are allowed to be slaughtered (which is only allowed if they have some sort of permanent blemish)? He may.

When Rav was going to Bavel, Rabbi Chiya said to Rebbe: My sister's son is going to Bavel. May he decide in matters of ritual law? He may. May he decide in matters of monetary law? He may. May he decide that first born animals are allowed to be slaughtered (which is only allowed if they have some sort of permanent blemish)? He may not.

...Rav himself would find an open space {a place where people transgressed the law} and fence it in {make the law more stringent}. For Rav was once in Tatelfush, and he heard a woman asking her number how much milk was necessary to cook a certain portion of meat. Rav said, "Do people here not know that cooking meat with milk is not allowed?" He therefore stayed some time and forbade them even to eat the udder.

Rav Yehuda said in the name of Rav: A man is forbidden to betrothe a woman until he will see her, for fear that he will then see something that is disgusting to him, which will cause him to go against the commandment, "love your neighbor as yourself."

Rav Yehuda said in the name of Shmuel: Just as it forbidden for one to leave the land of Israel for Bavel, so, too, is it forbidden to leave Bavel for other lands.