Question

Let R be a not necessarily commutative ring with identity. Let s be a nonzero element of r such that rs and sr are nonzero for all nonzero elements r in R; we say in this case that s is regular. Continuing to assume that s is a regular element of R,
Prove that the assignment \(r\mapsto rs\) for r in R, produces an injective abelian group homomorphism from R, regarded as an additive abelian group, to itself.
Please help

Answer 1

Define \(\phi : \mathbb R\rightarrow \mathbb R\\~~~~~~~r\mapsto rs\)