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I don't remember whether there was a special name for a commutative ring where every non-invertible element is a zero-divisor. And I also forgot the different ways in which a non-invertible element can be nasty. One obvious way to be nasty is to be a zero-divisor. So now

I'm looking for an example of a commutative ring which is not a subring of a commutative ring where every non-invertible element is a zero-divisor.

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    $\begingroup$ See also mathoverflow.net/questions/42647/…. $\endgroup$ – user26857 Nov 3 '14 at 9:26
  • $\begingroup$ Thanks for the link. So instead of "every non-invertible element is a zero-divisor", I could say "every regular element is a unit". Sweet, short and positive... $\endgroup$ – Thomas Klimpel Nov 3 '14 at 9:33
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Every commutative ring $R$ is a subring of its total ring of fractions, which is precisely the "natural" version of $R$ where every element is either a unit or a zero-divisor. Specifically, it is the localization of $R$ at the multiplicative subset $S\subset R$ consisting of the non-zero-divisors.

Thus, there is no such commutative ring $R$ with the property you're looking for.

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