The Multinomial Theorem in a Commutative Ring

In the setting of Finite Combinatorics: Notation, let $R$, with $+$, $\cdot$, $0$ and $1$, be a commutative ring. Let $m\in\mathbb{N}$, $x_{1},\dots,x_{m}\in R$ and $n\in\mathbb{N}_{0}$, and let $K=\{k\in\mathbb{N}_{0}^{m}:k_{1}+\dots+k_{m}=n\}$, with tuples as in Tuples in a Set: the Set of n-Tuples and Their Components (i).

(i) Multinomial theorem.

$$(x_{1}+\dots+x_{m})^{n}=\sum_{k\in K}\binom{k_{1}+\dots+k_{m}}{k_{1},\dots,k_{m}}x_{1}^{k_{1}}\cdots x_{m}^{k_{m}}.$$