Endomorphisms of the maximal ideal of a local ring

Question

Let $R$ be a commutative local ring with maximal ideal $\mathfrak{m}$. Is it true in general that $\text{Hom}_R(\mathfrak{m},\mathfrak{m})\cong \text{Hom}_R(\mathfrak{m}, R)$? What if the Krull dimension of $R$ is equal to one?

Answer 1

Let $R = \mathbb{Z}_2$, $\mathfrak{m} = 2\mathbb{Z}_2$. Then $x\mapsto x/2$ is an $R$-module homomorphism $\mathfrak{m}\to R$ whose image isn't contained in $\mathfrak{m}$.