homomorphism

A homomorphism is a structure-preserving mapping between two algebraic structures of the same type, e.g. two groups, two rings, or two vector spaces.

Formally,
a function $h : A \to B$ is a homomorphism if for each endowed relation $\circ_{i}$ on set $A$ which corresponds to $\circ_{i}^{'}$ on set $B$ ,

h (x \circ_{i} y) = h (x) \circ_{i}^{'} h (y)

In particular, the identity element of the first structure must be mapped to the identity element of the second structure.

Example: Ring Homomorphism

Let $R$ and $S$ be rings. Let $f : R \to S$ be a ring homomorphism.
Then for or all $a, b$ in $R$ ,

$f (a + b) = f (a) + f (b)$
$f (a b) = f (a) f (b)$
$f (1_{R}) = 1_{S}$