Gemara

(Mishnah): One who gives to his inn-keeper (dough to bake) must Ma'aser the dough he gives to her and the bread he receives, because she is suspected to switch it.

Avodah Zarah 22b: Rav Yosef taught that a widow may not keep a dog, or host a Talmid to dwell in her inn.

Question: Granted, she is suspected of sinning with the Talmid, for he is modest (if they sinned, he would not reveal the matter);

However, why do we forbid her to keep a dog?

Eruvin 53b -R. Yehoshua: Once I stayed at an inn, and my hostess served to me (bread and) beans on the first two days. I ate them, but I did not leave anything in the bowl. On the third day she ruined the beans with too much salt, and rebuked me for not leaving a remnant in the bowl (for me).

Rishonim

Rambam (Hilchos Isurei Bi'ah 22:16): A Talmid Chacham may not dwell in a Chatzer in which a widow lives, due to suspicion, unless his wife is with him.

Ra'avad: The Rambam went overboard. Chachamim said only that he may not lodge with her!

Magid Mishneh: Perhaps the Rambam holds that lodging, which is sporadic, is permitted, as long as they are not secluded. Eruvin 53b connotes like this. Also Chulin (6b) connotes like this. We forbid only dwelling, which is fixed, when they are in one Chatzer.

Migdal Oz: I say that the Gemara is even more stringent than the Rambam! In Kidushin (81a), we forbid seclusion with animals. Abaye and Rav Sheshes cleared animals away. All the more so, we are concerned for a widow!

Taz (EH 22:10): I say that the Rambam learned from the expression of 'Shera' (dwelling), which connotes a fixed residence. This is because there is no seclusion, just concern lest they come to seclusion. The Ra'avad learned from 'host', which means that she honors him and feeds him her food, the way people host guests. If he dwells in his own place and does not need her, this does not bring closeness. There is no question from R. Yehoshua. He ate and drank his own and paid normal rental to stay in an inn. This is her income with everyone!

Note: A support to say that he gave his food is that she rebuked him for not leaving a remnant in the bowl. If she gave her own food, why would she need him to leave a remnant for her?

Taz: It seems that Tosfos (Bava Metzi'a 71a DH Lo) disagrees. He says that we forbid her to keep a dog just to avoid slander, but really Yisrael are not suspected of bestiality and there is no Isur of seclusion. If so, the Isur to host a Talmid is also a mere stringency. If not, the Gemara taught the bigger Chidush first, that we are stringent more than letter of the law, and then it teaches that what letter of the law forbids. (This is unreasonable - PF.) Also, the Gemara equates them. It says 'granted, she is suspected of sinning with the Talmid. Why do we forbid her to keep a dog?' Therefore, I say that we need not harbor suspicion nowadays that Chachamim dwell in a Chatzer with a widow, and all the more so about Hagados Kelev (a comment suggests that this should say 'Halanah Levad' (mere lodging). Shulchan Aruch Rosh Pinah rejects this. Rather, it should say Hagdalas Kelev (raising a dog)) in a widow's house. One may rely on Tosfos.

Merkeves ha'Mishneh: The Targum of "Ki Shachan Alav he'Anan (Shemos 40:35)" is "Arei Shera Alohi Anana". The Targum of Shachan (dwelling) is always Shera. Achsnai in Hebrew is a guest. The Rambam's Perush is correct. He says that it is forbidden in the Chatzer, for the Gemara would not need to teach about the house, since it is forbidden due to seclusion.

Merumei Sadeh (22b DH v'Lo): A widow may not host a Talmid even if there is no concern for seclusion. The Gemara said that the Isur is on her, unlike the Rambam and Shulchan Aruch, who forbid a Talmid to stay with a widow. Had the Gemara said that he is forbidden, this would connote that it is forbidden even on occasion, like the Isur of seclusion. Really, it is forbidden only on a fixed basis, both according to the Rambam and Shulchan Aruch who forbid living in a Chatzer with a widow, and according to the Ra'avad, who forbids lodging in her house. Both permit occasionally. We find that R. Yehoshua lodged in a woman's inn. (Derech Eretz 4:3 says that she was a widow.) The Taz distinguishes between for free and for pay. This is wrong. Rather, we distinguish between occasionally and on a fixed basis. The concern is lest she accustom him and draw close to him. If we were concerned for him, we would forbid even once, like the Isur of seclusion. This was taught with the Isur for her to keep a dog. Also there, the concern is lest she draw close to it.

Mahariyo (55): A man asked if he may enter a widow and her son into his house to serve him. If no one else is in the house, there is an Isur of seclusion. A woman may not be secluded with two men on the road, for when one man needs to urinate, the other will be secluded with her. All the more so here, sometimes the son will go to the market or to stroll, and you will be secluded with his mother. We are concerned even for a short amount of time, i.e. enough to urinate, and all the more so here, the son will leave for large amounts of time! Even though here she is single, and the Isur of seclusion is only mid'Rabanan, we are more stringent for this than for seclusion with a Torah Ervah. The Gemara forbids a widow to house a Talmid in her inn. I.e. one may not live by a widow even without seclusion. We do not say so regarding a Torah Ervah. The Rambam forbids dwelling in a Chatzer in which a widow lives, even without seclusion, unless his wife is with him. All the more so one may not take her into his home! There, we must be concerned lest they come to sin! Also, since you are single, we must be more stringent than for others. In Kidushin 82a, we forbid a single man to teach children, because the mothers come occasionally, and we are concerned lest they come to sin. All the more so here is forbidden, for you invite her to be with you.

Me'iri (71a DH Ishah): One must distance himself from suspicion, ugly matters and everything similar. Therefore, Chachamim warned a widow not to take into her house any man, even a Talmid who came to learn Torah.

Poskim

Shulchan Aruch (EH 22:17): A Talmid Chacham may not dwell in a Chatzer in which a widow lives, due to suspicion, unless his wife is with him.

Chelkas Mechokek (19): The Ra'avad said that the Isur is only to lodge with her. The Magid Mishneh proved that one may lodge occasionally. The Isur is only to dwell permanently in the same Chatzer, even if it is not in the same house.

Beis Shmuel (18): The Taz forbids only if he eats her food, but permits if he pays her.

Otzar ha'Poskim (footnote 3): Apei Zutrei says that since there is suspicion, what is the source to say that it depends on whether he pays her? Rather, even if he pays, there is suspicion if he lodges according to the Ra'avad, or dwells in the Chatzer according to the Rambam. It seems that Apei Zutrei saw only the Beis Shmuel, for the Taz explained why it depends on whether he pays. He also asked why the Rema did not bring the Ra'avad's opinion, i.e. 'some say...'

Shai l'Morah (in Kovetz Meforshim in Shulchan Aruch Rosh Pinah, 7): This law is difficult, for Eliyahu dwelled with a widow (Melachim 1:17) and ate her food. Pirkei d'R. Eliezer expounds "Ki Basa Elai" to refer to Bi'ah.

Note: Hash-m commanded Eliyahu to do so, but one may ask why Hash-m would ask him to do something that arouses suspicion. I did not see that Pirkei d'R. Eliezer (33) explicitly expounds "Ki Basa Elai" to refer to Bi'ah. Perhaps Shai l'Mora learns like the Radal (below).

Radal (on Pirkei d'R. Eliezer 33:17): The Gemara forbids a Talmid, for he is modest and would not tell (if he sinned with her). She told Eliyahu 'since I am a widow and you are Ish Elokim, i.e. you are modest, there is concern lest people say that we had Bi'ah, and this is why my son died. Therefore, you must revive my son to save us from suspicion. Eliyahu told Hash-m that His command to dwell in her house caused this (therefore, it is proper for Him to revive him, and remove the suspicion).

Hagahos Maharam Berlin (5): Even though the Ra'avad forbids only in the same house, and Eliyahu was in the attic, this is considered like the house regarding suspicion.