Definition 2

Let A and B be two rings. A morphism, or homomorphism, of A into B is any mapping f of A into B satisfying the relations:

for all $x, y$ in A.

The composition of two ring homomorphisms is a ring homomorphism. Let A and B be two rings and $f$ a mapping of A into B; for $f$ to be an isomorphism, it is necessary and sufficient that it be a bijective homomorphism; in that case, $f^{-1}$ is a homomorphism of B into A. A homomorphism of a ring A into itself is called an endomorphism of A.

Let $f : A \to B$ be a ring homomorphism. The mapping $f$ is a homomorphism of the additive group of A into the additive group of B; in particular, $f(0) = 0$ and $f(-x) = -f(x)$ for all $x \in A$. The image under $f$ of an invertible element of A is an invertible element of B and $f$ induces a homomorphism of the multiplicative group of A into the multiplicative group of B.