The Quran gave women the right to divorce
“…If you fear that they are not able to keep within the limits of Allah, then there is no blame upon either of them concerning that the woman gives (افْتَدَتْ) a ransom / divorce compensation …” (al-Baqara [ 2] 229).
The tradition says that women don’t have the right to divorce. But Allah gave women the right to divorce in the Quran.
The word If’tada افْتَدَتْ in the verse at al-Baqara  229 is about the woman. In order to get a divorce, the woman goes to the authorities who previously practiced the marriage (al-Ahzab  37). However, the decision is the woman’s, not the authorities. The authorities only decide on the ransom (mahr). The authorities naturally try to save the marriage first (an-Nisa  35). But if the marriage can’t be saved, then they’ll do an investigation about the mahr. Mahr is something that the man gave to the woman at the marriage (money, jewellery, real estate).
The authorities decide whether the woman returns the whole mahr or just a little of it (al-Baqarah  229). After it is returned, the woman is divorced from her husband. She does not need to wait any period of time (iddah).