If a noetherian ring R is isomorphic to a ring S is S noetherian too?
I am pretty sure that it isn't but can't find a counterexample
Thanks!
2026-03-25 18:44:12.1774464252
Ring isomorphisms maintain noetherian property?
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It is indeed. Suppose $\phi:R\to S$ is an isomorphism. If $P_1\subset P_2\subset\cdots$ is an ascending chain of ideals in $S$, then $\phi^{-1}(P_1)\subset\phi^{-1}(P_2)\subset\cdots$ is an ascending chain of ideals in $R$, and $P_n=P_{n+1}$ if and only if $\phi^{-1}(P_n)=\phi^{-1}(P_{n+1})$.