Everything I've read and a number of etiquette resources state that the woman gets to keep the ring. The basis is that the ring is a gift from you to her and that it is wrong to ask for a gift back, no matter what the price was. That being said, I personally understand your point and don't see a problem with trying to get it back but I would keep it low key, especially if you want to remain social with her.
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A little rudeness and disrespect can elevate a meaningless interaction to a battle of wills and add drama to an otherwise dull day. Calvin
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