  The purpose of this short note is to prove that if $A$ and $B$ are unital
C*-algebras and $\phi : A \to B$ is a unital *-preserving ring homomorphism,
then $\phi$ is contractive; i.e., $\| \phi (a) \| \leq \| a \|$ for all $a \in
A$. (Note that we do not assume $\phi$ is linear.) We use this result to deduce
a number of corollaries as well as characterize the form of such unital
*-preserving ring homomorphisms. (This note may be of interest to
C*-algebraists as well as algebraists who study noncommutative rings and
algebras. It is meant to be accessible to a general mathematician and does not
require any prior knowledge of C*-algebras.)
