The Mishnah (3:5) teaches that if a nochri gave an Yisrael flour to prepare dough, then the resulting dough would be exempt from challah. In other words, even though the dough was prepared by an Yisrael, since the dough is owned by a Nochri it is exempt from challah. The Yerushalmi cites the Tosefta that teaches that in the opposite case, where the product was owned by an Yisrael but prepared by a Nochri, challah would need to be separated. Put simply, since the Torah uses the term "arisoteichem" -- "your dough" -- when referring to the obligation of separating challah, it is the ownership of the dough that is critical, not the person who prepared the dough.
This is indeed how the Shulchan Aruch rules (YD 330:1). The Shach cites the Derisha who explains that there was a practice that women would purchase dough for Nochri bakers and then separate challah from that dough. The Derisha explains that this practice is incorrect -- the dough was owned by a Nochri and therefore exempt from challah. The only way to rationalise this practice is if the women pre-paid. Then we can rely on the fact that on a biblical level ma'ot kanot. In other words, when they first handed over the money, it is considered as if they already purchased the dough and it belong to an Yisrael at the time of kneading. Alternatively, one could suggest that since this was the regular practice, when the Nochri prepared the dough for them, it is considered as if the ingredients were gifted to them in advance and the dough already belonged to an Yisrael.
Let us turn our attention now to the Rambam. The Rambam (Terumot 1:13) rules that if a Nochri formed an Yisrael's produce into a pile (digun) then, on a biblical level, it is exempt from terumot and maaserot.1 The Derech Emunah notes that many Rishonim disagree with the Rambam on this point and maintain that even if a nochri performed the diggun for produce owned by an Yisrael it would still be obligated in terumot and maaserot. Nevertheless, we shall try to understand the Rambam's position.
There are legal parallels between digun for produce and forming dough for the laws of challah. For example, these stages are considered gmar melecha such that one cannot snack from either prior to separating terumot and maaserot or challah. Are there parallels in this law in our Mishnah also?
The Derech Emunah (Bikkurim 8:52) notes that the Rambam would maintain the same position for challah also. In other words, if a Nochri prepared an Yisrael's dough it would be exempt for challah. Note that this is not like the Yerushalmi or Shulchan Arucha cited above. However, this is how the Tur rules (Hilchot Eretz Yisrael Challah 7) -- even though he forbids an Yisrael from doing so deliberately.
The Derech Emunah however cites the Ohr Zaruah (Challah 235) that differentiates between challah, and terumot and maaserot. For terumot the Torah uses the word "degancha". In other words, the Torah hinges the obligation on digun -- the act of forming the pile. For challah however, the Torah does not write "gilgulcha" -- "your forming the dough". It states "arisoteichem" -- "your dough". Consequently, while for terumah the act is important, for challah the material ownership of the dough is critical and not who formed the dough.
Indeed, the Chazon Ish (Sheviit 1:16) cites the Shittah Mekubetzet (Menachot 67a:1) who also brings this distinction. Importantly he adds that this is indeed the position of the Rambam.
1 Rabbinically it would still require separation to prevent this practice, as way of avoiding terumot and maaserot, from becoming widespread.)
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