Definition:Characteristic of Ring

From ProofWiki
Jump to: navigation, search

This page is about the characteristic of a ring, which subsumes such structures as integral domain, division ring and field. For other uses, see Definition:Characteristic.

Definition

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


Definition 1

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 $\operatorname{Char} \left({R}\right) = 0$.


Definition 2

Let $g: \Z \to R$ be the initial homomorphism, with $g \left({n}\right) = n \cdot 1_R$.

Let $\left({p}\right)$ be the principal ideal of $\left({\Z, +, \times}\right)$ generated by $p$.


The characteristic $\operatorname{Char} \left({R}\right)$ is the positive integer $p \in \Z_{\ge 0}$ such that $\left({p}\right)$ is the kernel of $g$.


Definition 3

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 defined as

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

Some others define the concept only on fields.


Also see