God made Eve from his side, not his foot or his head. They were companions, not master and subject.
Eve was deceived. Yet it was because of Adam's disobedience that death passed to all men. Evidently, he had more information about the Creation, since he spent a considerable period cataloging and naming the land animals. God placed Adam in the eastern (meaning earliest) garden before Eve existed. It was to Adam, that God gave the instructions not to eat the forbidden fruit. She knew the instruction since she explained it to the serpent. Thus she went against her husband who doubtless had warned her about the tree. She also gave fruit to her husband.
Perhaps it was because she went against the instructions of her husband, that God told her her desires will be to her husband and he will rule over you.
Before sin, they were innocent. They had no need for a ruler. All the world was theirs. No need for laws about property. The only law they had, other than the tree, was that they would be one, clinging to each other (The law of marriage).
Rulers, the Bible states, are in authority for our good. God did not intend for those in authority to lord it over others or to imagine that they are greater than those who are subject to them. Yet in an evil world, there is a need for a system of authority. However, the Bible tempers this with a command. We are to love our wives as Christ loved the church. I know of no man who can say I always obey this command perfectly.