\begin{definition}[Definition:Power of Element/Ring]
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 $n \in \Z_{\ge 0}$:
:$\forall n \in \Z_{\ge 0}: r^n = \begin {cases}
1_R & : n = 0 \\
r^{n - 1} \circ r & : n > 0
\end {cases}$
Category:Definitions/Ring Theory
\end{definition}