Answer to Question #212275 in Abstract Algebra for Nthabi

Question

let R be a ring in 1

1. let r be a ring with identity and let S be a subring of R containing the identity. prove that if u is a unit in S then u is a unit in R.

Answer 1

Let R be the ring with identity and S be the subring containing identity .

u is unit in S then there exists nonzero v in S such that uv=1

Now as S$\subset$ R, u,v are in R as well.

So, for u in R, there exists nonzero v in R so that uv=1.

Hence u is unit in R.