Definition:Characteristic of Ring/Definition 2

Definition

Let $\struct {R, +, \circ}$ 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 $\map g n = n \cdot 1_R$.

Let $\ideal p$ be the principal ideal of $\struct {\Z, +, \times}$ generated by $p$.

The characteristic $\Char R$ of $R$ is the positive integer $p \in \Z_{\ge 0}$ such that $\ideal p$ is the kernel of $g$.

Also see

• Equivalence of Definitions of Characteristic of Ring

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By Kernel of Ring Homomorphism is Ideal and Ring of Integers is Principal Ideal Domain, there exists a unique $p \in \Z_{\ge 0}$ such that $\ker g$ is the principal ideal $\ideal p$.

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $24$. The Division Algorithm