If $B$ is a (unital) subring of a unital local ring $A$, is $B$ still a local ring?
If not, under which assumption is it true?
2026-03-25 01:16:24.1774401384
Is a subring of a local ring still local?
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Suppose that for every element $c$ in $A$, we can find a unit $a \in A$ and an element $b \in B$ such that $ab = c$. Then it is easy to see that the map $$ I \mapsto I \cap B, $$ sending ideals of $A$ to ideals of $B$, is an order isomorphism w.r.t. inclusion.