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I don’t know if this concept already has a proper name in the commutative ring theory literature.
I have taken the liberty of making mild cosmetic adjustment to the wording in the first sentence.
Also I have hyperlinked the term zero divisor, throughout.
Replaced reference to “the” author of the nLab article by a more, how should I say, egalitarian formulation. (Sprinkled throughout the nLab are various phrases like “the author” or “I don’t know whether” or “I will come back to this later”, which I think generally should be discouraged, even though I have been guilty of doing this as well.)
Also clarified which ring of fractions (in the theorem before the examples section).
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