Gemara

(Beraisa - Aba Binyamin): I always exerted myself that my Tefilah should be near my bed.

This means that he would say Keri'as Shema and pray right after rising.

13b (Rav): When Rebbi teaches us, I do not see him interrupt to accept on himself Ol Malchus Shamayim at the time of Keri'as Shema!

R. Chiya: He does so when he passes his hand over his face.

14b: Rav would wash his hands, say Keri'as Shema, put on Tefilin and then pray.

He holds like R. Yehoshua ben Korchah, who says that we accept Ol Malchus Shamayim (Shema) before Ol Mitzvos (Tefilin).

Objection: R. Chiya bar Ashi said that often, Rav would wash his hands, bless, teach, put on Tefilin, and say Keri'as Shema!

Suggestion: He on Tefilin before Shema only if the time for Shema had not yet come.

Objection: If so, we would not learn anything from R. Chiya's testimony!

Answer: He teaches unlike the opinion that we do not recite Birkas ha'Torah for Mishnah.

Question: In any case, there is a contradiction in Rav's practice! (Above, we said that he said Shema before putting on Tefilin.)

Answer: The case was, the messenger delayed bringing Rav's Tefilin. (He needed to say Shema within the time, before his Tefilin came.)

Rishonim

Rosh (1:7): Aba Binyamin would pray right after rising. He would not work beforehand. Rashi says that also, he would not learn in between. This is astounding. What is his source for this?! Perhaps he discusses one who learns in his Beis Midrash, and he does not normally go to the Beis ha'Keneses. There is concern lest he be distracted in his learning, and the time for Tefilah and Keri'as Shema will pass.

R. Yonah (3a DH Al): Aba Binyamin discusses Shacharis. One should not do any Melachah before praying. He cannot refer to Ma'ariv (that it should be right before going to sleep), for one must pray Ma'ariv immediately at Tzeis ha'Kochavim, before eating and drinking. Rashi says that he would not learn after dawn before praying. The Tosafists say that there is no source to forbid this. Even Rashi forbids only when he learns alone, for he can pray and learn afterwards. Surely, one may teach others, since the time will pass. Merit of the Rabim is a great matter. Rebbi would teach, and when the time for Shema came, he recited Keri'as Shema and resumed teaching (13b). He prayed afterwards. Even though the beginning time for Tefilah arrived, he was not concerned lest he forget and the time will pass. Since he normally prays every day, he will put the matter to his heart and remember to pray.

Rashi (14b DH u'Masni): Rav taught before putting on Tefilin and saying Keri'as Shema. I.e. he engaged in doing a Mitzvah before accepting Ol Malchus Shomayim.

Tosfos (5b DH Ela): A proof that one may learn before Tefilah is from 14b. Rav would wash, bless, teach, and say Keri'as Shema when the time came.

Poskim

Shulchan Aruch (OC 89:6): One may not even learn before praying once the time comes.

Bach (8): What was Tosfos' question? Surely, before Zman Tefilah, i.e. before dawn, one may learn. One may even eat then, just he must stop at dawn. All the more so one may learn, just he stops when the time for Keri'as Shema and Tefilah come. Rav began learning before dawn. David rose at midnight and learned. Tosfos must explain that Aba Binyamin rose before dawn, for he would not transgress "A'irah Shachar." Even so, he was careful not to learn before praying. After eliminating and washing his hands, dawn came. Tosfos holds that he may learn until the time k'Vasikin (people who finish Shema and begin the Amidah at sunrise) if he prays alone, or in the Beis ha'Keneses if he prays with the Tzibur. Hagahos Maimoniyos (Hilchos Tefilah 6:5) says that Rashi and Tosfos argue only about between dawn and sunrise. People are lenient.

Mishnah Berurah (30): We forbid beginning to learn. If one began learning, even if he began after the time came, he need not interrupt even after the time comes, if he estimates that there will remain time to pray within the time. Some obligate interrupting for Keri'as Shema after the time comes, if he started after the time began. Some are lenient also about this, if there is no concern lest the time pass.

Mishnah Berurah (31): The time for Tefilah is from dawn. Before this, one may even begin, and then he need not interrupt later, unless he fears lest the time pass. Some permit beginning even after dawn, unless it is close to sunrise, which is the proper time to pray l'Chatchilah.

Kaf ha'Chayim (46): The Bach says that people are lenient to Tosfos to permit until close to sunrise. One must estimate how much time he will need to put on Tzitzis and Tefilin, and for the four parts of Tefilah that culminate with Keri'as Shema and the Amidah at sunrise. If time remains, he may learn. Even though the Mechaber rules like Rashi (who forbids from dawn), Torah after midnight and before Tefilah is very elevated. Even though it is greatest before dawn, also after dawn and before Tefilah begins he must learn, especially if he did not learn before dawn.

Kaf ha'Chayim (44): Shortly before Tefilah one may not take a child in his lap, lest the child soil his garment. Even if he washes it, it will not be as clean as before, or while he washes it he will miss Zman Tefilah or answering to Kaddish. Or, perhaps the child will cry when he wants to go, and due to his mercy on him he will neglect his Creator's honor and not go to the Beis ha'Keneses.

Kaf ha'Chayim (45): If a Chacham omitted Tefilah due to learning, even if he learned and taught the entire day, it is as if he did not learn that day. This is based on the Zohar. When one sleeps, his Neshamah sees the supreme light above, and he tastes death. His Neshamah does not return until he prays. The Yerushalmi says that even R. Shimon bar Yochai used to stop learning in order to pray.

Shulchan Aruch (ibid.): This refers to who often prays in his Beis Midrash, and he does not normally go to the Beis ha'Keneses. There is concern lest he be distracted in his learning and the time for Keri'as Shema and Tefilah will pass.

Mishnah Berurah (32): This is when he learns alone. If a Tzibur normally gathers, it is permitted. Surely they will remind him when the time comes.

Mishnah Berurah (33, citing Chayei Adam): Even if he has a fixed quota to learn, and there is not so much concern lest his learning extend, even so it is forbidden. Therefore, if one is on the road or lives in a settlement (where there is no Minyan), even in summer (when the time for Keri'as Shema and Tefilah extends very long), he should pray immediately. Even though there is a Mitzvah to pray when the Tzibur prays, since he may not learn before praying, it is better to pray immediately, lest he be idle from Torah. Nowadays that most travelers do not learn on the road, one should wait until the Tzibur prays. However, this can lead to missing the time for Keri'as Shema and sometimes even for Tefilah, so they should pray immediately upon rising before engaging in anything. All the more so, in big congregations in which they pray once it gets light out, this (immediately after rising) is when the Tzibur prays.

Mishnah Berurah (34): If one tells someone else who is not learning 'if I learn too long, remind me not to miss the time (for Keri'as Shema or Tefilah), it is permitted.

Shulchan Aruch (ibid.): If he teaches others, even if he does not normally go to the Beis ha'Keneses, it is permitted, since the time will pass. Merit of the Rabim is a great matter. If he will not teach them now, they will be Batel, and they will not be able to learn.

Gra (DH Hainu): We must say that Rashi forbids only one who learns alone, due to Tosfos' question from Rav. Clearly he taught even after the time came, for the Gemara said "v'Chi Teima..."

Damesek Eliezer: Even though the Gemara concluded that the Shali'ach came late, the words "v'Chi Teima..." show that the suggestion (that the time had not yet come) is not true.

Gra: The question was not (just) that he taught before putting on Tefilin. The time for Tefilin and Shema are the same. Do not say that Rav holds k'Vasikin (and therefore he delays saying Shema until just before sunrise, and then prays immediately). He put on Tefilin between Keri'as Shema and Tefilah. This shows that he does not hold k'Vasikin, according to the Hava Amina!

Shulchan Aruch (ibid.): If he normally goes to the Beis ha'Keneses, it is permitted.

Kaf ha'Chayim (49): The Chayei Adam says that this is only if someone calls people to come to pray. The Poskim did not specify. This connotes that it suffices if he goes, even if no one calls.