\begin{definition}[Definition:Characteristic of Ring/Definition 1]
Let $\struct {R, +, \circ}$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$.
For a natural number $n \in \N$, let $n \cdot x$ be defined as the power of $x$ in the context of the additive group $\struct {R, +}$:
:$n \cdot x = \begin {cases}
0_R & : n = 0 \\
\paren {\paren {n - 1} \cdot x} + x & : n > 0
\end {cases}$
The '''characteristic''' $\Char R$ of $R$ is the smallest $n \in \N_{>0}$ such that $n \cdot 1_R = 0_R$.
If there is no such $n$, then $\Char R = 0$.
\end{definition}