I'm looking for a reference to the following corollary, which comes as a by-product of a more general result.

Corollary. Let $R$ be a Dedekind-finite unital ring (either commutative or not), and let $k \in \mathbf N^+$. Then the monoid ring $R[\mathbb N^k]$ has infinitely many pairwise non-associate irreducible elements.

Notes. (i) A unital ring is called Dedekind-finite provided that $xy = 1_R$ for some $x, y \in R$ only if $yx = 1_R$. (ii) $\mathbb N^k$ is the monoid $(\mathbf N^k, +)$. So in particular, if $R$ is commutative, then $R[\mathbb N^k]$ is a ring of polynomials in $k$ variables with coefficients in $R$.