Ring Homomorphisms

A mapping \(f : R \rightarrow S\) where \(R, S\) are rings is a ring homomorphism if

  1. \(f(1) = 1\)

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

\[ \array { f(x+y)&=&f(x) +f(y) \\ f(x\cdot y) &=& f(x)\cdot f(y) } \]

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 \(f\), \(f(R) = \{f(x)|x\in R\}\) 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.

Ben Lynn blynn@cs.stanford.edu 💡