Gemara

(R. Ami): We do not teach Divrei Torah to Nochrim - "Lo Osah Chen l'Chol Goy u'Mishpatim Bal Yeda'um".

Bava Kama 38a (Rav Yosef): "...Ro'ah va'Yater Goyim" - Hash-m saw that they were not keeping their seven Mitzvos, and released them from their Mitzvos.

Objection: Do they gain from their sins?!

Answer (Mar brei d'Ravina): No - it means, even if they fulfil their Mitzvos, they are not rewarded for them.

Question (Beraisa - R. Meir): "(The Mitzvos) that a man will do and live in them" - it doesn't say that Yisraelim will do, rather that a man will do!

This teaches that a Nochri who engages in Torah is (rewarded) like a Kohen Gadol.

Answer: They are not rewarded like one who is commanded, rather like one who is not commanded.

(Beraisa): Romi sent two officers to learn the Torah; Chachamim taught them. They acknowledged the truth of every point, except for one.

The officers: How can it be that a Yisrael's ox that gores a Nochri's ox is exempt, but a Nochri's ox (even a Tam) that gores a Yisrael's ox pays full damage?

If one is liable only for "Shor Re'ehu (of his fellowman)", a Nochri should be exempt when his ox gored a Yisrael's ox;

If we do not expound "Shor Re'ehu" like this, a Yisrael should be liable when his ox gored a Nochri's ox!

Sanhedrin 59a (R. Yochanan): A Nochri who learns Torah is Chayav Misah. "Torah...Morashah Kehilas Yakov", it is an inheritance for Yisrael, not for Nochrim.

Question: Why isn't this included among his seven Mitzvos?

Answer #1: It is included in theft;

Answer #2: According to the opinion that expounds as if said Me'orasah (Torah is Mekudeshes to Yisrael), it is included in Arayos.

Question (Beraisa - R. Meir): A Nochri who engages in Torah is like a Kohen Gadol. "Asher Ya'aseh Osam ha'Adam va'Chai" includes Nochrim.

Answer: He is rewarded for learning his seven Mitzvos (but liable for anything else).

We never find something permitted to a Yisrael and forbidden to a Nochri.

Shmos Rabah (30): Akilas told Andriyanus (Kaiser) that he wanted to convert to learn Torah. He explained that an Arel cannot learn - "Lo Osah Chen l'Chol Goy..."

A Nochri was walking outside a Beis ha'Keneses and heard about the Begadim of the Kohen Gadol. He requested to be converted on condition to be appointed Kohen Gadol; Hillel did so, and told him to learn about kingship (Kehunah).

The Nochri learned "Veha'Zar ha'Karev Yumas", and realized that he cannot be Kohen Gadol.

Rishonim

Rambam (Hilchos Melachim 8:10): Moshe bequeathed the Torah and Mitzvos only to Yisrael - "Morashah Kehilas Yakov". It is also for anyone of another Umah who wants to convert.

Rambam (10:9): A Nochri who learns Torah is Chayav Misah. He may learn only his seven Mitzvos. If he learned Torah, kept Shabbos or made new Mitzvos, we hit and punish him and tell him that he is Chayav Misah, but we do not kill him.

Kesef Mishneh: Since it is not listed among his seven Mitzvos, he is not killed (b'Yedei Adam) for this. It seems that it is only an Asmachta to say that it is theft or Bi'as Na'arah Me'urasah.

Chasam Sofer (Chulin 33a DH v'Ayen): The Rambam distinguishes between a Nochri and a Ben No'ach who accepted not to serve idolatry. The latter may keep any Mitzvos he wants; we may teach him Torah.

Note: The Rambam (10:10) says that we do not stop a Ben No'ach who wants to keep other Mitzvos. In Perush ha'Mishnayos (Terumos 3:9) he says that if a Nochri did any Mitzvah he gets a little reward.

Question: The Gemara says that we never find something permitted to a Yisrael and forbidden to a Nochri. A Nochri may not learn Torah or keep Shabbos!

Answer #1 (Tosfos Sanhedrin 59a DH Leika): We do not discuss things that are Mitzvos for Yisrael.

Answer #2 (Lechem Mishneh Melachim 10:9): Mid'Oraisa these are permitted to Nochrim. The Isur and Chiyuv Misah are mid'Rabanan.

Answer #3 (Lechem Mishneh 9:13): The Rambam holds that it is an argument whether or not we find things permitted to a Yisrael and forbidden to a Nochri. The Halachah is, we find such things.

Teshuvas Rambam (149 (Blau), brought in Likutim in Frankel Rambam): When Yisraelim are in control, we do not allow a Ben No'ach to learn Torah until he converts. One may teach Mitzvos to Notzrim to draw them to our creed. One may not teach Yishmaelim, for they think that our Torah is not from Shomayim, and when it opposes what they fabricated, they will explain it according to their dogma. This can lead simpleminded Yisraelim to err. The Arelim (Notzrim) believe in our written Torah, just they explain it wrong. It is possible that they will retract.

Question: Why do we need a source to forbid teaching Torah to Nochrim? A Nochri is forbidden to learn, so one who teaches him transgresses "V'Lifnei Iver Lo Siten Michshol"! We cannot say that this applies to their seven Mitzvos, for it is a Mitzvah to teach them their Mitzvos!

Answer (Tosfos Chagigah 13a DH Ein): If there is someone else they could learn from, e.g. another Nochri, Lifnei Iver does not apply.

Question: Why did Chachamim teach Torah to Nochrim? This is forbidden!

Answer (Tosfos Bava Kama 38a DH Kar'u): The ruling authority forced them to teach them. We need not be Moser Nefesh to avoid teaching Nochrim. Alternatively, the Nochrim expressed desire to convert, like it says in Sifri (v'Zos ha'Berachah 344, which describes the episode slightly differently).

Yam Shel Shlomo (Bava Kama 4:9): We learn from the episode and Tosfos that one may not teach Torah to Nochrim for the sake of Shalom with the Malchus, and surely not for the sake of pay. We see that people who do so cause heresy. The Torah commanded not to write Torah she'Ba'al Peh to avoid this problem. We also learn that one may not falsify Torah, even if there is danger. One must be Moser Nefesh to avoid this. Chachamim knew that telling them that a Nochri is liable but a Yisrael is exempt could spark harsh decrees. Nevertheless, it was forbidden to falsify this; falsifying Torah is like denying it. The 70 who translated the Torah for Ptolemy changed several matters; this was through Ru'ach ha'Kodesh (all made exactly the same changes). Also, they changed only the wording, not the meaning.

Maharsha (Shabbos 31a DH Omar Lei Mikra): We do not accept converts who seek wealth. Surely, the same applies to one who seeks to wear Bigdei Kohen Gadol! We must say that Hillel did not convert him until the Nochri learned that he cannot be Kohen Gadol. Even though one may not teach Torah to Nochrim, since he wanted to convert it is permitted.

Rebuttal (R. Akiva Eiger Teshuvah 41): Tosfos (Yevamos 24b DH Lo and 109b DH Ra'ah) explains that Hillel converted him because he knew that he would eventually convert Lishmah. Presumably, Tosfos did not explain like the Maharsha because one may not teach to one who seeks to convert!

Note: All agree that we may teach some things to one who comes to convert! We inform him of some light Mitzvos, some severe Mitzvos... we tell him that after converting he will be Chayav Kares for eating Chelev and stoned for Chilul Shabbos (Yevamos 47a). R. Akiva Eiger means that one may not teach more than is necessary for conversion.

Mishneh Halachos (2:42 DH Od Hevi and DH Od Yesh): We can say that Hillel relied on hi Chachmah that the Nochrim he taught would eventually convert. There are two opinions in the Yerushalmi about whether or not Antoninus converted. According to the opinion that he did not, why was Rebbi allowed to teach him?

Poskim

Gra (YD 246:22): If a woman learns Torah, she gets reward. She is no worse than a Nochri!

Shiyurei Berachah (on 246:6): It is a great sin even to teach a Nochri Aleph-Beis. Some say that R. Ami said that we are not Moser Divrei Torah to Nochrim to teach that we may not sell to them Sifrei Kodesh. This is not a Diyuk, but the law is true.