Endomorphisms of the maximal ideal of a local ring 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?
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 $\mathfrak{m}$.
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 $\mathfrak{m}$.