Definition:Characteristic of Ring/Definition 1

Definition
Let $\left({R, +, \circ}\right)$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $n \cdot x$ be defined as in Definition:Power of Element.

The characteristic of a ring with unity $R$ (written $\operatorname{Char} \left({R}\right)$ or $\operatorname{char} \left({R}\right)$) is the smallest $n \in \Z, n > 0$ such that $n \cdot 1_R = 0_R$.

If there is no such $n$, then $\operatorname{Char} \left({R}\right) = 0$.

Also see

 * Equivalence of Definitions of Characteristic of Ring