ring homomorphism RingHomomorphism 2001-10-19 00:34:55 2006-10-22 09:15:34 Definition unital ring isomorphism ring epimorphism ring monomorphism homomorphism isomorphism epimorphism monomprhism \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{graphicx} \usepackage{xypic} 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.