Example

(1) Let A be a ring. It is immediately seen that the mapping $n \mapsto n \cdot 1$ of $\mathbf{Z}$ into A is the unique homomorphism of $\mathbf{Z}$ into A. In particular, the identity mapping of $\mathbf{Z}$ is the unique endomorphism of the ring $\mathbf{Z}$.

In particular, take A to be the endomorphism ring of the additive group $\mathbf{Z}$ (no. 3, Example VI). The mapping $n \mapsto n \cdot 1$ of $\mathbf{Z}$ into A is an isomorphism of $\mathbf{Z}$ onto A by the very construction of multiplication in $\mathbf{Z}$ (§ 2, no. 6).

(2) Let $a$ be an invertible element of a ring A. The mapping $x \mapsto axa^{-1}$ is an endomorphism of A for

It is bijective, for the relation $x' = axa^{-1}$ is equivalent to $x = a^{-1}x'a$. It is therefore an automorphism of the ring A, called the inner automorphism associated with $a$.