# Definition:Characteristic of Ring/Definition 1

## Definition

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 $\Char R = 0$.