Definition:Characteristic of Ring/Definition 3

Definition
Let $\left({R, +, \circ}\right)$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$.

The characteristic of $R$, denoted $\operatorname{Char} \left({R}\right)$, is defined as follows.

Let $p$ be the order of $1_R$ in the additive group $\left({R, +}\right)$ of $\left({R, +, \circ}\right)$.

If $p \in \Z_{>0}$, then $\operatorname{Char} \left({R}\right) := p$.

If $1_R$ is of infinite order, then $\operatorname{Char} \left({R}\right) := 0$.

Also see

 * Equivalence of Definitions of Characteristic of Ring