\begin{definition}[Definition:Norm/Ring/Submultiplicative]
Let $\struct {R, +, \circ}$ be a ring whose zero is denoted $0_R$.
A '''submultiplicative norm''' on $R$ is a mapping from $R$ to the non-negative reals:
:$\norm {\,\cdot\,}: R \to \R_{\ge 0}$
satisfying the '''(ring) submultiplicative norm axioms''':
{{:Axiom:Submultiplicative Norm Axioms}}
\end{definition}