If a ring has a 1 (i.e. a ring has a multiplicative inverse), does this imply that the subring has a 1 as well?
Thank you
2026-04-06 22:32:16.1775514736
Subrings and multiplicative identities
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If your definition of "subring $S$ of $R$" includes verbiage saying "and it has to have the same identity as $R$" then trivially yes.
But some authors do not require that, and some don't require identities at all. So, it just depends on what definitions you are using.
An abelian subgroup that is closed under multiplication could have no identity element, or it could have an identity distinct from that of $R$, or it could have the same identity as $R$. Which one of these you call "a subring" is up to the definitions you've settled on.