I was solving tasks in a book and there are two tasks where I have no idea how to solve them:
1) Let $R$ be commutative ring, $f: M\rightarrow N$ - isomorphism of $R$-modules. Prove isomorphism of modules $\operatorname{End}_R(M)$ and $\operatorname{End}_R(N)$.
@Crostul proposed to observe this map $F: \phi \mapsto f^{-1}\circ\phi\circ f$. It's easily to check that this is bijection. We need to show that:
a) $F(\phi + \psi) = F(\phi) + F(\psi)$
$F(\phi + \psi)(x) = f^{-1}\circ(\phi + \psi)\circ f(x) = f^{-1}\circ(\phi(f(x)) +\psi(f(x))) \\= f^{-1}\circ\phi\circ f(x) + f^{-1}\circ\psi\circ f(x)$.
All equations are correct because $f$ - isomorphism of modules and $\phi$ - homomorphism (endomorphism) between modules
b)$F(a\phi) = aF(\phi)$ where $a\in R$. It's also obvious.
But I did not understood why $R$ must be commutative? Can you show me situation where it's necessary?