Is the localization of a direct product of two rings at a maximal (or prime) ideal identified with a localization of one of them? I would appreciate for any detailed answer.
2026-03-28 10:16:10.1774692970
Localization of a direct product
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First prove the following: if $S_i\subset R_i$ is a multiplicative set, then $$(S_1\times S_2)^{-1}(R_1\times R_2)\simeq S_1^{-1}R_1\times S_2^{-1}R_2.$$
A maximal ideal of $R_1\times R_2$ is of the form $M_1\times R_2$ or $R_1\times M_2$ with $M_i$ maximal in $R_i$. To localize $R_1\times R_2$ at $M_1\times R_2$, consider the multiplicative set $$(R_1\times R_2)\setminus (M_1\times R_2)=(R_1\setminus M_1)\times R_2,$$ use the previous isomorphism and recall that $S^{-1}R=0$ if $0\in S$.