The Yerushalmi [Kiddushin 1-1] says:
[We will explain what is necessary for our understanding as we go along].
Now R' Chiya [at the end of the passage] first says that goyim don't have geirushin. Then he talks about a goy who divorces his wife and she marries another man and then gets divorced from him too. Then she converts together with her first husband. In that case she may marry him and there is no איסור of not taking back an ex-wife after she marries another man. This implies that the Ran is correct that goyim may divorce and when it says that they don't [like R' Chiya said at the outset] it means בכתב - with a document. Otherwise, how does it say that she divorced her husband and married another man? That isn't divorce - that is adultery [if a goy can't get divorced]?! It must be that they can't get divorced with a document but they can in a different way. According to this line of thought, when R' Yuda ben Pazi said that they don't have geirushin ["או שאין להן גירושין"], he meant בכתב only. And that is what the Korban Ha-edah explains there on the daf. But if so, it is hard to understand what it means when it says:
"או שאין להן גירושין או ששניהן מגרשין זה את זה"
"Either they don't have divorce or they divorce each other".
This seems to mean that if they don't have divorce [in writing], then only the man can perform the divorce but if they do have divorce [in writing], then they may divorce each other. What does the fact that a divorce is in writing have to do with the fact that they can both do it? And what exactly is the dilemma between the two positions of them either not having or yes having divorce?
The Shiyarei Korban writes as follows:
או שאין להן גירושין וכו'. כתב הרמב"ם פ"ט מה' מלכים הלכה ח'. ומאימתי תהיה אשת חבירו כגרושה שלנו משיוציאנה מביתו וישלחנה לעצמה או משתצא היא מרשותו ותלך לה שאין להם גירושין בכתב ואין הדבר תלוי בו לבד אלא כל זמן שירצה הוא או היא לפרוש זה מזה פורשין ע"כ. וקשה איך תפס דברי שניהן שאין להם גירושין ושניהם מגרשין זה את זה ור' יוחנן לא קאמר הכי אלא אחת משתיהן או שאין להן גירושין או ששניהן מגרשין זה את זה.... א"כ דברי הרמב"ם צ"ע:
He quotes the Rambam who says that they don't have geirushin in writing but they can both decide to leave and end it. Asked the S.K. - How did the Rambam pasken both sides, that they don't have geirushin in writing and they can both decide to end it, when the Yerushalmi said that it is either-or? Either they don't have geirushin in writing [in which case it is implicit that only the husband can do it] or they do [in which case they can both break it off] but not both and the Rambam paskens both?!
Also, the Yerushalmi has a doubt about the question of whether they can both initiate the divorce, so how did the Rambam know to pasken with simplicity that they can?
In Breishis Rabba [י"ח ח] it says:
ומניין שאין להם גירושין? ר' בר סימון ור"ח בשם ר' יוחנן אמר שאין להם גירושין או ששניהם מגרשין זה את זה א"ר יוחנן אשתו מגרשתו ונותנת לו דופורין.
Here in the Medrash, the girsa is not "או או" presenting two options between not having geirushin or them both being able to divorce but rather seems to be more decisive: They don't have geirushin at all [period!] or they can both do it [not "either or". Definitely option "a" or maybe option "b" as well]. What does that mean?
What is the meaning of אשתו מגרשתו ונותנת לו דופרין? The עץ יוסף explains that it means "כתובה כפולה" - double. What does THAT mean??
The Radbaz explains that the reason that "מגרשין זה את זה", both divorce each other, is that since they are not בתורת קידושין - they don't have formal marriage, they are also not בתורת גירושין - they don't need a formal divorce [where he gives her a document of separation but rather each side can just walk away]. So it appears from the language of the Rambam who says:
The Radbaz explains that the reason that "מגרשין זה את זה", both divorce each other, is that since they are not בתורת קידושין - they don't have formal marriage, they are also not בתורת גירושין - they don't need a formal divorce [where he gives her a document of separation but rather each side can just walk away]. So it appears from the language of the Rambam who says:
"שאין להם גירושין בכתב ואין הדבר תלוי בו לבד"
"They don't have divorce in writing and it is not dependent on him alone".
We have to understand! He already said earlier that that the divorce can be effected by either party so why does he repeat himself? We are compelled to say that he is offering the rationale for the halacha. SINCE they don't have a written divorce, it is not dependent on him alone. OK. But that is a non-sequitur! What does one have to do with the other? It must mean that the law of divorcing with a document is predicated on the קנין of the husband on the wife that was activated via the קידושין. The purpose of the document is to nullify the קנין, as the Rambam writes [Geirushin 1-1] that the idea of the כתב [document] is that "שגרשה והסירה מקנינו" - she is "expelled" and displaced from his acquisition of her. When it comes to בני נח, all there is no קנין but rather a "מצב" of אישות, a situation where they are living in matrimony through their common volition. This "מצב" can be dissolved when either party has a will to end this connection. Therefore, she can end it just as well as he can!
This is also indicated by what the Rambam says:
"ומאימתי תחזור להיתרה"
When does she return to her state of being permitted [for relations]?
Why does the Rambam say "תחזור" which implies that she is returning to her original state of being permitted. But this is not a "returning" since she became forbidden forever but is rather is a COMPLETELY NEW state of היתר? From here as well we see that that there is no eternal קנין on the woman that requires הפקעה [dissolution] at which point she receives a new lease on freedom. But rather, from the get-go, there was only an איסור as long as there is a מצב of אישות and they are living together as man and wife. If so, vis a vis the time AFTER the dissolution of their union she was never forbidden in the first place and she indeed "returns" to her original היתר.
See Rashi in Kiddushin [מ"א: ד"ה לאו] that an עבד is doesn't have גירושין because he doesn't have קידושין. Now this is puzzling. It is PASHUT that when an עבד כנעני is in a monogamous, conjugal union with a שפחה כנענית anyone who has relations with her is in BIG TROUBLE because she is considered a married woman. So what does Rashi mean when he says that an עבד doesn't have קידושין [see Minchas Chinuch 35]?
The explanation is as we said: What Rashi means is that an עבד doesn't have a proper, formal קנין in a woman and all he has is a "מצב" of "אישות" and therefore he has no need for a proper גט to undo the קנין קידושין.
According to this, the דין of מגרשין זה את זה - each can divorce the other, is because there is no proper קנין. As Rashi there explains, since it says "והיו לבשר אחד" - they will be as one flesh, there is a notion of equality between the couple. A מציאות of unity. All that has to happen for this unity to dissolve is for one of the sides to decide that they are no longer interested in this union. At that point, it ends.
That is the meaning of the gemara [Sanhedrin 82a] that says that there is no אישות for בני נח. The concept of אישות implies בעלות, as Rashi says [Rus 1-3] איש נעמי - לפי שהיה איש ושולט עליה. The man has a certain amount of בעלות over the wife. אישות denotes that the male is the more dominant figure [and thus he is the one who performs the act of קידושין and gives the גט - notwithstanding the Ritva we saw in a recent article who says that when a man gets married he essentially becomes a slave to the woman because of his many formidable obligations towards her]. But when it comes to בני נח, there is a complete equality as expressed in the pasuk "they will be as one flesh". That is not authentic "אישות" but a union of equal one-ness.
[עפ"י תורת מו"ר הגאון הגדול רבי ד"י מן זצ"ל]