Let $\displaystyle R$ be a ring with unity and let $\displaystyle End(\langle R,+\rangle)$ be the ring of endomorphisms of $\displaystyle \langle R,+\rangle$. Let $\displaystyle a\in R$, and let $\displaystyle \lambda_a : R \rightarrow R$ be given by $\displaystyle \lambda_a(x)=ax,\forall x\in R$.

a) Show that $\displaystyle \lambda_a$ is an endomorphism of $\displaystyle \langle R,+\rangle$

b) Show that $\displaystyle R'=\{\lambda_a|a\in R\}$ is a subring of $\displaystyle End(\langle R,+\rangle)$

c) Prove the analogue of Cayley's theorem for $\displaystyle R$ by showing that $\displaystyle R'$ of (b) is isomorphic to $\displaystyle R$