The Image of the Natural Numbers with Zero in a Commutative Ring Respects Zero, One, Sums, Products, Differences, Powers, and Finite Sums and Products

In the setting of The Natural Numbers and the Natural Numbers with Zero: Arithmetic, Order, Induction and Recursion, let $R$, with $+$, $\cdot$, $0$ and $1$, be a commutative ring, let $m,n\in\mathbb{N}_{0}$, and let $m_{R}$ and $n_{R}$ be their images in $R$. Let $-$ in $R$ be as in Negatives, Differences, Reciprocals and Quotients §negative, $n-m$ for $m\le n$ in $\mathbb{N}_{0}$ as in The Difference of Two Natural Numbers with Zero §difference, powers in $\mathbb{N}_{0}$ and in $R$ as in Powers with Exponents in the Natural Numbers with Zero §power and Powers with Exponents in the Natural Numbers with Zero §zero, and finite sums and products in $\mathbb{N}_{0}$ and in $R$ as in Sums and Products over a Finite Set and over an Interval §operation and Sums and Products over a Finite Set and over an Interval §empty; these apply in $R$ because $+$ and $\cdot$ are associative and commutative with neutral elements $0$ and $1$ by Commutative Rings §ring, on both sides by commutativity, and in $\mathbb{N}_{0}$ by The Natural Numbers and the Natural Numbers with Zero: Arithmetic, Order, Induction and Recursion §laws.

¶ $0_{R}$ is the zero of $R$, and $1_{R}$ is the unit of $R$.

¶ $(m+n)_{R}=m_{R}+n_{R}$.

¶ $(mn)_{R}=m_{R}\,n_{R}$.

¶ If $m\le n$, then $(n-m)_{R}=n_{R}-m_{R}$.

¶ $(m^{k})_{R}=(m_{R})^{k}$ for every $k\in\mathbb{N}_{0}$.

¶ For every finite set $A$ and every map $f:A\to\mathbb{N}_{0}$, $\big(\sum_{a\in A}f(a)\big)_{R}=\sum_{a\in A}f(a)_{R}$ and $\big(\prod_{a\in A}f(a)\big)_{R}=\prod_{a\in A}f(a)_{R}$.