Definition:Characteristic of Ring

This page is about Characteristic in the context of Ring Theory. For other uses, see Characteristic.

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 $\Char R = 0$.

Let $g: \Z \to R$ be the initial homomorphism, with $\map g n = n \cdot 1_R$.

The characteristic $\Char R$ of $R$ is the positive integer $p \in \Z_{\ge 0}$ such that $\ideal p$ is the kernel of $g$.

The characteristic of $R$, denoted $\Char R$, is defined as follows.

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

If $p \in \Z_{>0}$, then $\Char R := p$.

If $1_R$ is of infinite order, then $\Char R := 0$.

Some authors insist that the characteristic is defined on integral domains only.

Some others define the concept only on fields.

Some sources use $\infty$ rather than $0$ for rings which have no characteristic defined.

Some sources use $\map {\operatorname {char} } R$ to denote the characteristic of a ring $R$.