Ring Homomorphisms
A mapping $f : R \rightarrow S$ where $R, S$ are rings is a ring homomorphism if

$f(1) = 1$

For all $x,y\in R$ we have
It is easily verified that if $f$ is a ring homomorphism, then:

$f(0)=0$

$f(x) = f(x)$ for all $x \in R$

The image of $f$, $f(R) = \{f(x)x\in R\}$ is a subring of $S$
Also it is clear that the composition of ring homomorphisms is also a ring homomorphism.
An isomorphism is a bijective homomorphism. If $f:R\rightarrow S$ is an isomorphism we write $R \cong S$. Note that isomorphism is an equivalence relation.