The following comes as a by-product of a more abstract result, and I'm essentially looking for a reference to it (or to something more general than it).

Corollary. Let $R$ be a non-trivial Dedekind-finite unital ring (either commutative or not), $k$ a positive integer, and $\Gamma$ a non-trivial submonoid of $(\mathbf N^k, +)$. Then the monoid ring $R[\Gamma]$ 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) If $\Gamma = (\mathbf N^k, +)$ and $R$ is commutative, then $R[\Gamma]$ is just the usual ring of polynomials in $k$ variables $X_1, \ldots, X_k$ with coefficients in $R$.