Definition:Characteristic of Ring

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

Alternative Definition
Alternatively, it can be defined as follows.

Let $g: \Z \to R$ be the homomorphism defined as $\forall n \in \Z: g \left({n}\right) = n \cdot 1_R$.

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

Then $\operatorname{Char} \left({R}\right)$ is the integer $p \in \Z_+$ such that $\left({p}\right)$ is the kernel of $g$.

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

Some others define the concept only on fields.