13B10-RingHomomorphism.tex

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\begin{document}
Let $R$ and $S$ be rings. A \emph{ring homomorphism} is a function $f: R \longrightarrow S$ such that:
\begin{itemize}
\item $f(a+b) = f(a)+f(b)$ for all $a,b \in R$
\item $f(a\cdot b) = f(a) \cdot f(b)$ for all $a,b \in R$
\end{itemize}

A \emph{ring isomorphism} is a ring homomorphism which is a bijection. A \emph{ring monomorphism} (respectively, \emph{ring epimorphism}) is a ring homomorphism which is an injection (respectively, surjection).

When working in a context in which all rings have a multiplicative identity, one also requires that $f(1_R) = 1_S$. Ring homomorphisms which satisfy this property are called \emph{unital} ring homomorphisms.
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