Units map to units - Imported Knowlpedia Content

Units map to units: Let $\varphi:R\to S$ be a unital ring homomorphism. If $u\in R$ is a unit, then $\varphi(u)\in S$ is a unit and

\varphi(u^{-1})=\varphi(u)^{-1}.

Equivalently, $\varphi$ restricts to a group homomorphism $R^\times\to S^\times$ on unit groups.

Remarks

For unital rings, any ring homomorphism induces a homomorphism of the group of units by sending each unit to its image. In particular, any ring isomorphism identifies the unit groups of the two rings.