Powers and Images of Natural Numbers in a Commutative Ring Lie in the Ring

In the setting of Commutative Rings, Fields and Ordered Fields: Standard Notation, let $R$, with $+$, $\cdot$, $0$ and $1$, be a commutative ring.

¶ For every $x\in R$ and every $n\in\mathbb{N}_{0}$, the power $x^{n}$ for the multiplication $\cdot$, as in Commutative Rings, Fields and Ordered Fields: Standard Notation §rings and Powers with Exponents in the Natural Numbers with Zero §zero, lies in $R$: $x^{n}\in R$.

¶ For every $n\in\mathbb{N}_{0}$, the image $n_{R}$ of $n$ in $R$ lies in $R$: $n_{R}\in R$.