Gemara

65b (R. Chanina bar Yosef, and R. Yochanan): If a Nochri [in the Chatzer was away, and] returned on Shabbos, we may rent [his Reshus].

(Rav Asi): We may not rent.

Rav Sheshes: R. Elazar found the first opinion difficult, for it is unlike R. Elazar's Rebbi, Shmuel!

(Shmuel): Whenever [Yisraelim] forbid each other but cannot make an Eruv, they cannot be Mevatel.

Suggestion: This is when there is a Nochri.

If the Nochri was there before Shabbos, they should have rented from him then! Rather, the Nochri returned on Shabbos, and Shmuel taught that they forbid each other, they cannot make an Eruv, and they cannot be Mevatel!

67a Question (Rav Chisda): If a Nochri died on Shabbos, what is the law?

We do not ask according to those who permit renting from a Nochri on Shabbos. They permit two things, so all the more so they permit one!

70b (Beraisa): The general rule is, whatever was forbidden for part of Shabbos is forbidden for all of Shabbos, except for Bitul Reshus.

'The general rule is' includes when a Nochri died on Shabbos.

Rishonim

Rambam (Hilchos Eruvin 2:12): One may rent from a Nochri on Shabbos, for rental is like Bitul Reshus. It is not a Vadai rental. It is a mere Heker (reminder).

Rambam (13): If two Yisre'elim and a Nochri live in one Chatzer, and they rented from the Nochri on Shabbos, one of them is Mevatel is Reshus to the other, and he is permitted. Similarly, if the Nochri died on Shabbos, one Yisrael is Mevatel to the other, and it is permitted to carry.

Rosh (6:9): Rashi says that after renting from the Nochri on Shabbos, the Yisre'elim are Mevatel Reshus to one of them, and that Yisrael is permitted. The entire Sugya proves that both rental and Bitul are needed. Even though they were Me'arev, rental is not enough. Once the Nochri comes, the Eruv is Batel. Rental does not return the Eruv to be valid. We say 'this is unlike Shmuel, who says that they forbid each other, they cannot make an Eruv, and they cannot be Mevatel!' We say (67a) that those who permit renting from a Nochri on Shabbos permit two things, i.e. renting and Bitul.

Rosh: In Shabbos (101b) we say that boats tied to each other may carry from one to the other. If they became untied, they are forbidden. If they were tied again, the Heter returns. Here is different, for from the beginning the Eruv was not destined to last for all of Shabbos, for the Nochri will return. There, the boats were not destined to come apart. If one member of the Chatzer forgot to be Me'arev, his Eruv is not Batel. Rather, he is Mevatel his Reshus to them, and they are permitted. There is different, for all of them could have been Me'arev from before Shabbos. Here 'once Shabbos was permitted, it was permitted' does not apply. Do not say that here we need Bitul because they were not Me'arev, but if they were Me'arev, rental works without Bitul, for the Eruv returns to its initial Kashrus. Surely they were able to be Me'arev, for the Halachah follows R. Yehudah, that a Nochri forbids only when he is here. Since they are able to be Me'arev, what is the source to say that they were not Me'arev, and rented and were Mevatel? Perhaps they were Me'arev, and only rented! Rather, even if they were Me'arev, the Eruv is Batel when the Nochri comes.

Rosh: Why was rental needed? Bitul should suffice! After they are Mevatel to one of them, it is as if one Yisrael lives with the Nochri, like on 63b! There, Rava concluded that Bitul does not help, for if so, [they will always rely on Bitul, and] Eruv will be forgotten. That is when the Nochri does not want to rent. Here, the Nochri agrees to rent. They can be Me'arev on another Shabbos., so Eruv will not be forgotten. I answer that these Amora'im hold like R. Meir, who forbids an individual with a Nochri. R. Yochanan holds that the custom is like R. Eliezer ben Yakov, but we do not rule like him. However, we hold like R. Eliezer ben Yakov, so Bitul alone helps without rental. Alternatively, since if the Nochri would refuse to rent, Bitul would not help [lest Eruv be forgotten], we do not distinguish between whether or not he wants. His words should not affect Bitul, i.e. it helps only if he says 'I want.' Therefore, we always require both.

Rosh: R. Chananel rules like R. Yochanan, for the Halachah always follows him against Shmuel, and like the lenient opinion in Eruvin. The Ri rules like Shmuel, for a Beraisa (70b) supports him. If a Nochri died on Shabbos, Bitul does not help. We say (67a) that if a Nochri died on Shabbos, R. Yochanan, who permits two things (renting and Bitul), surely permits one! This shows that the Beraisa cannot be like R. Yochanan. It is difficult to say that the Beraisa includes when a Nochri died on Shabbos, i.e. that Eruv without Bitul does not permit, but Bitul permits. We compare this to Nochrim who made a Mechitzah on Shabbos, which is [invalid] even through Bitul. Even if R. Yochanan can establish the general rule of the Beraisa in another way, the Stam Gemara did not. This shows that this is the Halachah. Maharam rejected the proof. We challenged Shmuel from the Beraisa, therefore the Gemara answered according to Shmuel. Also the Rif rules like R. Yochanan.

Poskim

Shulchan Aruch (OC 383:1): If a Nochri lives with two Yisre'elim, and the Nochri is not in his house, he does not forbid (see 371:1). They are Me'arev, and they are permitted.

Mishnah Berurah (2): The Nochri and his entire family went before Shabbos to another city.

Mishnah Berurah (3): In 371:1, some permit only if he went more than one day's journey away, and some permit even if he was closer than this.

Kaf ha'Chayim (5): Ru'ach Chayim says that the measure of a day's journey (40 Mil - PF) did not change, even nowadays that there are fast steamboats. Misgeres Zahav on Kitzur Shulchan Aruch says that some disagree.

Shulchan Aruch (ibid.): If the Nochri came on Shabbos, he forbids and the Eruv is Batel.

Magen Avraham (1): Even if the Nochri was far away, he can walk during the week and come on Shabbos, unlike Olas Shabbos. All the more so, if boats were not tied before Shabbos, the Eruv is Batel, for they were not destined to be tied. Why does the Rosh permit this (1:23)? Also, 'all the more so if he died' connotes that even if the Nochri was here before Shabbos, and they were Me'arev, and he died on Shabbos or they rented from him, the Eruv is Batel, and they must be Mevatel, like Tosfos says. The Eruv is not rejuvenated. If so, why do we say (374:1) that it is rejuvenated, even if [two Chatzeros were Me'arev together, and the opening between them was sealed, so the Eruv was Batel, and] it opened on Shabbos?

Mishnah Berurah (4): Even if he went more than one day's journey away, he must have started returning before Shabbos, and at the start of Shabbos he was within one day's journey, and he was destined to come on Shabbos. Therefore, we do not say 'once it was permitted, it is permitted.'

Kaf ha'Chayim (7): If on Erev Shabbos it seemed clear that he will not come, and he reconsidered, Poskim argue about this. It is good to be stringent.

Shulchan Aruch (ibid.): However, they may rent from him on Shabbos, and then one of them is Mevatel is Reshus to the other, and he is permitted.

Beis Yosef (DH Goy): Tosfos, the Rosh, and Semag say that the Ri rules like Shmuel. It seems that Semak agrees. The Rif, Rosh and Rambam rule like R. Yochanan. It seems that the Rosh holds like his latter answer, that we always require rental and Bitul. Perhaps this is why he put it last, even though Tosfos put it first.

Mishnah Berurah (6): They may rent on Shabbos, for it is unlike business. It is a mere Heker to permit carrying.

Mishnah Berurah (7): The one to whom they were Mevatel (Ploni) is permitted. The Mevatel is forbidden, unless Ploni was later Mevatel to him.

Shulchan Aruch (ibid.): All the more so, if the Nochri died on Shabbos, one Yisrael is Mevatel to the other, and he is permitted.

Beis Yosef (DH u'Mah): This is clear according to R. Yochanan, whom we rule like.

Taz (2): Tosfos [and the Rosh] said that renting from the Nochri does not make the Eruv valid again, since it was destined to be Batel. I derive that when the Nochri died, one Yisrael is Mevatel to another, i.e. when they did not rent from him beforehand. Had they rented before Shabbos and made an Eruv, and when he died they rented from his heir, they would not need Bitul. This is like boats, which were not destined to come apart. He was not destined to die on Shabbos! All the more so, this applies if he died before Shabbos.

Mishnah Berurah (8): When we need rental and Bitul, they help. All the more so when we need only Bitul, it helps! This is even if they were not Me'arev before Shabbos, for the Nochri forbade them. If they rented from him before Shabbos and made an Eruv, when he dies the Eruv is intact, even before they rent from his heirs. Since the beginning was permitted, all of Shabbos is permitted.

Sha'ar ha'Tziyun (9): This is like Tosefes Shabbos, Ma'amar Mordechai and the Pri Megadim, unlike the Taz.

Kaf ha'Chayim (13): If the Nochri was Goses before Shabbos, since most Gosesim die, the Eruv was destined to be Batel.