End$_A(A)\cong A$

Question

I am wondering the following question.

Let $A$ be a ring. Is it always true that End$_A(A)\cong A$. Is true when $A$ is semisimple?

I tried to give an isomorphism $\phi$ between End$_A(A)$ and $A$. $\phi(f)=f(1)$, but I don't know if I am right.

Thanks for your help.

Answer 1

Yes and no: you have to specify the side of the module or if your ring is commutative or not to get a clear statement and answer.

For any ring with identity,

$End(R_R)\cong R$ and $End(_RR)\cong R^{op}$

using the obvious map that you're suggesting.