Commutative algebra

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nah this one will live forever :P

The trick to producing these kinds of counterexamples is to keep $U$ disjoint from $U^*$. If they have any elements in common then on one side of the isomorphism, the image of each $u$ goes to a unit, but on the other side, one of them goes to zero; that's what gets you the zero ring.

Instead, if $U$ is a multiplicatively closed set, then $U^*=\{v\in R: uv=0\}$ is an ideal, and $R[U^{-1}] \cong (R/U^*)[\pi(U)]$, where $\pi$ is the quotient map.

you're never around :P

62 days so far :P

Integral domain for the weekend: condensed domain