Definition:Characteristic of Ring/Definition 2

Definition
Let $\left({R, +, \circ}\right)$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $g: \Z \to R$ be the initial homomorphism, with $g \left({n}\right) = n \cdot 1_R$.

Let $\ker g$ be its kernel.

By Ideal of Ring of Integers has Unique Positive Generator, there exists a unique $p \in \Z_{\geq 0}$ such that $\ker g$ is the principal ideal $(p)$.

The characteristic $\operatorname{Char} \left({R}\right)$ is the integer $p$.

Also see

 * Equivalence of Definitions of Characteristic of Ring