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 $1_R$.
Definition 1
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 $\Char R = 0$.
Definition 2
Let $g: \Z \to R$ be the initial homomorphism, with $\map g n = n \cdot 1_R$.
Let $\ideal p$ be the principal ideal of $\struct {\Z, +, \times}$ generated by $p$.
The characteristic $\Char R$ of $R$ is the positive integer $p \in \Z_{\ge 0}$ such that $\ideal p$ is the kernel of $g$.
Definition 3
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 $\Char R := 0$.
Also defined as
Some authors insist that the characteristic is defined on integral domains only.
Some others define the concept only on fields.
Also denoted as
Some sources use $\map {\operatorname {char} } R$ to denote the characteristic.
Also see
- Equivalence of Definitions of Characteristic of Ring
- Characteristic of Division Ring is Zero or Prime
- Results about characteristics of rings can be found here.