tolkiens legendarium - If Galadriel had claimed the One Ring, would Sauron have become her servant, or would have Galadriel become his servant?

This question was inspired by the Andres F answer to another question. In his answer he stated

"we are told that if she ever took the One Ring, she would become way more powerful than Sauron ever was."

So, it makes me wonder: If Galadriel had claimed the One Ring, would Sauron have become her servant, or would have Galadriel become his servant?

Answer

Tolkien commented on that subject in the letter 236: Tolkien's letters

Of the others only Gandalf might be expected to master him – being an emissary of the Powers and a creature of the same order, an immortal spirit taking a visible physical form. In the 'Mirror of Galadriel', 1381, it appears that Galadriel conceived of herself as capable of wielding the Ring and supplanting the Dark Lord. If so, so also were the other guardians of the Three, especially Elrond. But this is another matter. It was part of the essential deceit of the Ring to fill minds with imaginations of supreme power. But this the Great had well considered and had rejected, as is seen in Elrond's words at the Council. Galadriel's rejection of the temptation was founded upon previous thought and resolve. In any case Elrond or Galadriel would have proceeded in the policy now adopted by Sauron: they would have built up an empire with great and absolutely subservient generals and armies and engines of war, until they could challenge Sauron and destroy him by force. Confrontation of Sauron alone, unaided, self to self was not contemplated. One can imagine the scene in which Gandalf, say, was placed in such a position. It would be a delicate balance. On one side the true allegiance of the Ring to Sauron; on the other superior strength because Sauron was not actually in possession, and perhaps also because he was weakened by long corruption and expenditure of will in dominating inferiors. If Gandalf proved the victor, the result would have been for Sauron the same as the destruction of the Ring; for him it would have been destroyed, taken from him for ever. But the Ring and all its works would have endured. It would have been the master in the end.

So we see that:

1. Even with the One Ring, Galadriel would not match Sauron one on one



2. She might be deluding herself when dreaming of overthrowing Sauron with the One

To add, Gandalf (Olorin) was of the same stock as Sauron, a Maiar, and essentially of stronger will than even any of the First Born. Gandalf would have been the likely successor of the ring, and sagely resisted the temptation in Book 1 of LOTR. Frodo, excellent keeper that he is, seeks to unload it upon the two greatest forces on two legs in Middle Earth, and both resist, recognizing his earnestness and the futility of acceptance.