Equivalence of Definitions of Characteristic of Ring

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Theorem

Let $\struct {R, +, \circ}$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$.

The following definitions of the concept of Characteristic of Ring are equivalent:

Definition 1

Let $n \cdot x$ be defined as in Definition:Power of Element.

The characteristic $\Char R$ of $R$ is the smallest $n \in \Z, n > 0$ such that $n \cdot 1_R = 0_R$.

If there is no such $n$, then $\Char R = 0$.

Definition 2

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 $g$.

Definition 3

The characteristic of $R$, denoted $\Char R$, is defined as follows.

Let $p$ be the order of $1_R$ in the additive group $\struct {R, +}$ of $\struct {R, +, \circ}$.

If $p \in \Z_{>0}$, then $\Char R := p$.

If $1_R$ is of infinite order, then $\Char R := 0$.


Proof