Definition:Power of Element/Ring

Definition

Let $\struct {R, +, \circ}$ be a ring.

Let $r \in R$.

Let $n \in \Z_{>0}$ be the set of strictly positive integers.

The $n$th power of $r$ in $R$ is defined as the $n$th power of $r$ with respect to the semigroup $\struct {R, \circ}$:

$\forall n \in \Z_{>0}: r^n = \begin {cases} r & : n = 1 \\ r^{n - 1} \circ r & : n > 1 \end {cases}$

If $R$ is a ring with unity where $1_R$ is that unity, the definition extends to $n \in \Z_{\ge 0}$:

$\forall n \in \Z_{\ge 0}: r^n = \begin {cases} 1_R & : n = 0 \\ r^{n - 1} \circ r & : n > 0 \end {cases}$