For an arbitrary ID (integral domain) $R$ with subring $S$, assume that $S$ has an unity. Then does $R$ have a unity too? If not, please provide a counter-example.
2026-03-27 12:12:16.1774613536
Does every ID with subring which has a unity have an unity?
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If $S$ is nontrivial (that is, has a nonzero identity $e$) then yes.
A nonzero idempotent in a domain must act as the identity for the domain.
To see this, just examine what $e(ex-x)$ and $(xe-x)e$ must be, where $x$ is an arbitrary element of the domain.
This came up under slightly different circumstances here.