Commutative Algebra

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

1. $f(1)=1$

2. For all $x,y\in R$ we have

It is easily verified that if $f$ is a ring homomorphism, then:

1. $f(0)=0$

2. $f(-x)=-f(x)$ for all $x\in R$

3. The image of $f$, $f(R)=\{f(x)\mid x\in R\}$ 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 $f:R\to S$ is an isomorphism we write $R\cong S$. Note that isomorphism is an equivalence relation.