Endomorphism ring

Remarks

$\End_R(M)$ is a ring with zero map as zero and the identity map $\id_M$ as unit element. For an abelian group $A$, viewed as a $\Z$-module, one writes $\End(A) = \End_\Z(A)$. Example: $\End_\Z(\Z/n\Z) \cong \Z/n\Z$. The check: an endomorphism $f$ is determined by $f(1) = a$, since $f(k) = ka$, and every $a$ gives one; composition corresponds to multiplication of the values $a$. Non-example: the map $x \mapsto x + 1$ on $\Z$ is not in $\End_\Z(\Z)$, since it does not preserve addition: it sends $0$ to $1$. For a field $K$, choosing a basis gives $\End_K(K^n) \cong M_n(K)$. For the ring $R$ as a left module over itself, $\End_R(R) \cong R^{\mathrm{op}}$, the opposite ring, through $f \mapsto f(1)$.