I am trying to come up with an example of a ring whose total quotient ring is not a field. I know that if $R$ is a domain, then every total quotient ring has to be a field, however in the general case I feel like there should be a counterexample.
2026-04-07 08:22:08.1775550128
Is there a ring whose total ring of fractions is not a field?
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As noted in the comments; the total ring of fractions of a ring $R$ is a ring containing $R$ as a subring. So if $R$ is not a domain then its total ring of fractions is not a field.
The smallest example of such a ring is $\Bbb{Z}/4\Bbb{Z}$. It is not a domain, and it is its own total ring of fractions, which is clearly not a field.