Example of a commutative ring which is not a subring of a commutative ring where every non-invertible element is a zero-divisor

Question

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.

Answer 1

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.