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 $1_R$.

Let $n \in \N$ denote a natural number. Let $n \cdot x$ be defined as in power of $x$ in the context of the additive group $\struct {R, +}$:


 * $n \cdot x = \begin {cases}

0_R & : n = 0 \\ \paren {n - 1} a + a & : 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 $\Char R = 0$.

Also see

 * Equivalence of Definitions of Characteristic of Ring