Definition:Characteristic of Ring

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 Power of an Element.

The characteristic of a ring with unity $$R$$ (written $$\operatorname{Char} \left({R}\right)$$) is the smallest $$n \in \mathbb{Z}$$ such that $$n \cdot 1_R = 0_R$$.

If there is no such $$n$$, then $$\operatorname{Char} \left({R}\right) = 0$$.

Alternatively, it can be defined as follows.

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

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

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

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