Ring Homomorphisms
A mapping \(f : R \rightarrow S\) where \(R, S\) are rings is a ring homomorphism if
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\(f(1) = 1\)
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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:
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\(f(0)=0\)
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\(f(-x) = -f(x)\) for all \(x \in R\)
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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.