# 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 $\left({p}\right)$ 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 $g$.

## Also see

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

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 24$