Gemara

(Mishnah): (Normally, one must redeem a Peter Chamor (a firstborn donkey, by giving its value or a Seh (goat or sheep) to a Kohen), or do Arifah (break its neck).) In the following cases, a Peter Chamor is exempt (from redemption or Arifah):

One bought the fetus in a Nochri's donkey;

One sold the fetus in his donkey to a Nochri, even though this is forbidden;

One was a partner with a Nochri in a donkey;

One was Mekabel from a Nochri (to take care of his donkey, and share the offspring), or he gave his donkey to a Nochri b'Kabalah.

We learn from "b'Yisrael" -- not of others.

3a - Question: (According to Chachamim,) how much must a Nochri own to exempt it?

Answer #1 (Rav Huna): Even if he owns an ear it is exempt.

Question (Rav Nachman): Even if the ear would be cut off and given to the Nochri, the remainder would have Kedushas Bechor!

Answer #2 (Rav Chisda): If removal of the Nochri's share would make it Nevelah, it is exempt.

Answer #3 (Rava): If removal of the Nochri's share would make it Treifah, it is exempt.

Rava and Rav Chisda argue about whether or not a Treifah can live. Rava holds that it cannot. Rav Chisda holds that it can. Therefore, the Nochri's share is not essential unless its removal would make it Nevelah.

Rabanan: Rav Chisda and Rava do not argue with Rav Huna. They say how much a Nochri must own of the mother to exempt. Rav Huna says how much he must own of the fetus to exempt.

Objection (Rav Papa): Presumably, you say that any (Nochri) ownership in the fetus exempts, because this is not "Kol Bechor";

If so, any ownership also in the mother should exempt, because this is not "v'Chol Miknecha Tizachar"!

Rather, there is no distinction between the mother and the fetus.

Question (against all three answers - Mar bar Rav Ashi): Why is this different than Nefalim (non-viable offspring)? They get Kedushas Bechor, even though they have no life at all!

"(Kedushas Bechor applies to) Peter Sheger Behemah." This is a fetus that was born prematurely (alternatively - a miscarriage).

Answer: There, the entire fetus is proper for Kedushas Bechor. This is "bi'Vehemah Kol Bechor." Here, there is a mixture of Chulin (the Nochri's share). The verse does not apply.

(R. Asi citing R. Yochanan): If removal of the Nochri's share would make even a minor Mum (blemish), it is exempt.

Rav Mari bar Rachel had a flock. He used to be Makneh (transfer ownership of) the ears to a Nochri, yet he would forbid working (with the Bechoros) and would give them to Kohanim. His flock died out.

Question: Why did his flock die out?

Answer #1: It was a punishment for uprooting Kedushas Bechor.

Answer #2: Rav Mari caused a Takalah. He knew how to be Makneh to a Nochri, but onlookers, who did not know the proper way, would be Makneh in an invalid way.

13a (Mishnah): In the following cases, a cow's first calf does not get Kedushas Bechor:

One bought the fetus in a Nochri's cow;

One sold the fetus in his cow to a Nochri, even though this is forbidden;

One was a partner with a Nochri in a cow;

One was Mekabel from a Nochri (to take care of his cow, and share the offspring), or he gave his cow to a Nochri b'Kabalah.

53a Suggestion: Chachamim should enact that people sell the ear to a Nochri to uproot Kedushas Bechor!

Answer: Since one can do like Rav Yehudah, there is no need to totally uproot Kedushas Bechor:

(Rav Yehudah): One may blemish a Bechor before it leaves the womb.

Rishonim

Hilchos Bechoros (4:1): Bechorah applies to animals of partners. "Bekarcha v'Tzoncha" excludes partnership with a Nochri. If a Nochri was a partner in a cow or the fetus, even if he owned one part in 1000 in the mother or child, this exempts from Bechorah. If one of them owned one limb, e.g. the fore or hind leg, if removal of that limb would make the animal a Ba'al Mum, it is exempt. If one could cut off the Nochri's limb and it would not be disqualified, Bechorah applies to it.

Rosh (1:2): The Halachah follows Rav Huna, for R. Yochanan holds like him. Also Rav Mari holds like him. He was punished only for uprooting the Kedushah (but not because they were really Bechoros and he caused Kohanim to stumble). The Stam Gemara (53a) suggested being Makneh (transferring ownership of) the ears to a Nochri to uproot the Kedushah. In Pesachim (6a) we conclude that animals of Arnona (the king collects a tithe from them) are exempt from Bechorah, even though one could give money in place of animals.

Rosh (2): People might have erred to think that Rav Mari was Makneh animals through mere speech. Even though he was punished, nowadays people are Makneh the ear. He was Makneh the ear of the fetus itself, and uprooted from the fetus itself. The Gemara connotes like this, for it says 'he was Makneh their ears, and forbade shearing and working with them.' This connotes the subject of these is the same, i.e. the fetuses. There is no punishment for being Makneh the ear of the mother.

Question: If there is no problem to be Makneh the ear of the mother, why did the Gemara (53a) say that it is better to blemish the Bechor?

Answer (Rosh): Since there is concern for Takalah, it is better to enact like Rav Yehudah without uprooting, Chachamim did want to enact in a way that uproots the (monetary benefit of the) Kohen. (I.e. Rav Mari voluntarily can the animals to a Kohen, but according to letter of the law, he could have kept them - PF.) Alternatively, we can say that in the days of Rav Yehudah people were experts to blemish the Bechor before it is born, therefore it was forbidden to uproot. Nowadays we are not experts to blemish the Bechor before it is born, therefore it is better to uproot than to come to Takalah. Rav Mari was punished because he knew to do like Rav Yehudah.

Rosh: R. Tam says that Rav Mari used to be Makneh the mother in order that the buyer acquire the ear of the fetus. People erred, and thought that he was Makneh (directly) the ear of the fetus, which is Davar she'Lo Ba l'Olam (it is not yet in the world; the buyer does not acquire). If one is Makneh the ear of the mother, there is no room for mistake. Rashi says that Rav Mari knew to take money from the Nochri, since a Nochri acquires only through money. People thought that he did not take money. According to R. Tam, mid'Oraisa money acquires (when Yisre'elim sell Metaltelim to each other), so it does not acquire for Nochrim. To evade a Safek, the Nochri must also do Meshichah. It is best that the Nochri pay a Perutah and acquire the place where the animal stands, and acquire its ear. It is good to do so, to avoid Takalah with a Bechor. One should not do so regarding a donkey, for one can avoid Takalah through redemption or breaking its neck.

Poskim

Shulchan Aruch (YD 320:3): Partnership of a Nochri exempts (from Bechorah of a Tahor animal). If a Nochri was a partner in a cow or the fetus, even if he owned one part in 1000 in the mother or child, this exempts from Bechorah. Therefore, if one accepted an animal from a Nochri to raise it and share the children, or if a Nochri accepted an animal from a Yisrael under such an agreement, they are exempt from Bechorah, for it says "Peter Rechem bi'Vnei Yisrael" - it must be totally a Yisrael's.

Beis Yosef (DH u'Mah she'Chasav v'Afilu): Here, Tosfos rules like Rav Huna, like the Rosh. However, in Avodah Zarah (71a DH Rav), Tosfos is stringent and says that one should be Makneh from the mother or fetus a part whose removal makes the animal Terefah, e.g. the head or lungs. However, the Nochri need not pay the full value of the head or lungs. It suffices for him to pay a Perutah, even if it is worth much more. We are not concerned for Bitul Mekach, for the Yisrael resolves to sell in order to exempt from Bechorah.

Question (Prishah 4): The Beis Yosef brought the opinion of Tosfos here and the Rosh, and of Tosfos in Avodah Zarah (and said that it is good to be stringent like Tosfos in Avodah Zarah - the Magihah). (Note: the Beis Yosef cited Tosfos in Avodah Zarah, who says that one should be stringent. The Beis Yosef did not explicitly say that one should be stringent.) Why is this different than Matanos Kehunah (foreleg, jaw and stomach)? The Beis Yosef (YD 61) said that the Nochri must own part of the entire animal. If he is a partner only in the head, this exempts only from (the Matanah that comes from the head, i.e.) the jaw. The same applies to the foreleg and stomach. It seems that Bechor is different, because its Kedushah applies equally to the entire animal. If the Nochri owns anything, this is a part that Kedushas Bechor applies to. Therefore, it is exempt. Matanos apply only to the foreleg, jaw and stomach. Therefore, a share in other parts of the animal does not exempt, and a share in one of the Matanos does not exempt from the others. However, if he owns a part of the entire animal, even one part in 1000, he has a share in all of the Matanos.

Taz (3): The Drishah's question is not difficult. Regarding Bechor, it says "Peter Rechem b'Yisrael", therefore it must be entirely the Yisrael's. The same applies to the fine for stealing and selling or slaughtering a Seh or cattle. We learn from "u'Machru", which connotes selling all of it (Bava Kama 78b). If any vital part was omitted, it (the entire sale) is void. Regarding Matanos, there is no verse (teaching that it depends on total ownership). Rather, whatever part the Nochri owns a share in, only that part is exempt.

Shulchan Aruch (321:20): Partnership of a Nochri exempts (from Bechorah in a donkey). The same laws of partnership of a Nochri in a Tahor animal apply to a Tamei animal. However, it is forbidden to make a Nochri a partner (in the donkey or the fetus) in order to uproot the Kedushah, since one can redeem the Bechor or break its neck.

Beis Yosef (DH u'Mah, citing the Rosh): We may uproot Kedushas Bechor of Tahor animals, for we have no alternative. Regarding a Peter Chamor, there are alternatives (without risk of Takalah).