We'll have a machloket (strife) between Beit Shammai and Beit Hillel here regarding Yibum. If a man dies with no children, his wife, or widow, is obligated to do Yibum—to marry the brother of the deceased man. If the deceased man doesn't want to marry her, he does chalitza, the removal of the shoe, and then she's free to marry somebody else. Generally, a brother is not allowed to marry the former wife of his brother; she would be considered an erva, a prohibited marriage. When it comes to Yibum, the Torah makes an exception, and a man is allowed to marry his brother's widow. But if the widow was an erva to him in a different way—for example, if she was a man's daughter—then he would not be allowed to do Yibum with her. But if there was another brother who was not an erva to her—for example, if this man we were talking about, whose daughter she was, happened to be the deceased brother's wife—and there was another brother to whom this widow was a niece, he would be obligated to do Yibum or chalitza with her.
Here we'll talk about a man who dies with no children and has more than one widow. He has a few different widows, but one of them is an erva to the living brother. Obviously, the erva woman, or the erva widow herself, cannot do Yibum. But what about the co-wives of that erva? She has other wives. Are they also exempt from Yibum and chalitza because she is, or not? The Mishnah says that in this situation, Beit Shammai would allow the co-wives of an erva to the brothers. They say that the brother is allowed to do Yibum with the erva's co-wife. Let's say you have a brother—you have the man here. One brother cannot do Yibum with his own daughter. Let's say his own daughter was married to his deceased brother. That's an erva. But what about if she had co-wives? There were other wives married to the man who passed away. So are they allowed to do Yibum? Is that other wife, who is not his daughter, allowed to do Yibum with this man? Beit Shammai's answer is matirin: he's allowed to. Even if he's the only living brother, he's allowed to marry the co-wife of his erva. Let's say it's his daughter's co-wife; he would be allowed to marry her. Beit Hillel says, “osrin”: the co-wife is forbidden to the brother. If he's the only living relative, then they would be free to marry other people without chalitza. Totally patur (exempt). No need for Yibum or chalitza.
There are more consequences to this machloket. Chaltzu, they performed chalitza. If one of the co-wives of the erva did chalitza with the living brother, Beit Shammai say poslin min ha-kehunah: they disqualify her from the priesthood. According to Beit Shammai, they would say the chalitza was meaningful. It worked, and therefore that co-wife is a chalutzah, someone who did chalitza. She would be prohibited from marrying into the kehunah. She couldn't marry a Kohen (priest). Beit Hillel say, “makhshirin.” Beit Hillel would say, “No, it's a meaningless chalitza.” The chalitza was nothing because there was no need for it; the woman was free to marry anybody even without the chalitza. So, therefore, she's not a chalutzah, and she would be allowed to marry a Kohen.
But if one of the co-wives was taken by the brother in Yibum and then he died, Beit Shammai say, “makhshirin.” Beit Shammai say she is now eligible to marry a Kohen because the Yibum was for sure allowed. It was a normal case of Yibum. She married the man, and eventually he died. So now she's a widow. A widow is allowed to marry a Kohen. But Beit Hillel say, “No, she's now disqualified from marrying a Kohen.” The Yibum was not allowed. So the woman had relations with a man she was not allowed to have relations with because he was the brother of her husband, and she wasn't allowed to be with him. And it wasn't a case of Yibum. So, therefore, she's now a zonah, and she's not allowed to marry a Kohen going forward.
When we look at Beit Shammai and Beit Hillel, we may think there was strife between them and that they were split into two separate communities. But we'll say here: even though Beit Hillel said certain women were disqualified from marriage and Beit Shammai said they were eligible for marriage, so they were seemingly problematic to each other halachically, we would think that the families on each side of the machloket would stay away from each other. But we said they did not. We said they did not. We didn't ban Beit Shammai from marrying a Kohen from Beit Hillel, and Beit Shammai did not hold back from marrying women from the school of Beit Hillel, and Beit Hillel did not hold back from marrying people from the school of Beit Shammai. They were always open and transparent with each other about what the other side would hold as forbidden. Shows how the Talmudic they bring more peace and tranquility into the world. They held it was okay, but they knew that the other opinion said it was a problem, so they made sure to inform them. Therefore, those people would not marry from Beit Shammai to Beit Hillel or from Beit Hillel to Beit Shammai. But in general, they would make sure to keep proper records and inform each other, and therefore they would normally be eligible to marry each other.
With all the pure items and tamei (impure) items that one side said were pure and the other side said were tamei, each side wouldn't hold back from preparing pure foods on utensils they borrowed from each other. They would always tell them, “By the way, this utensil is okay for me, but for you it's a problem, so don't take it.” They were so respectful of the other side that they would inform each other about all the halachot (laws) that the other side needed, even though they disagreed and didn't hold by them themselves.