Several different generalizations of finite factorization domains (i.e., integral domains where every nonzero nonunit has only finitely many divisors up to associates) have been defined for commutative rings with zero divisors. We study these notions in the context of commutative monoid rings with zero divisors, utilizing semigroup theory to simultaneously generalize and extend many past results about "finite factorization" properties in commutative polynomial rings. Along the way, we expand upon the general theory of factorization in commutative rings with zero divisors, providing new characterizations and results about several kinds of "finite factorization rings."
[This is the preprint of our paper with the same title that is now published in Communications in Algebra. This is the version before peer review. The published version has no changes in its results, but the presentation was changed a little bit and some more examples were added, so the numbering on the results is different in the final version. I can provide a link to get an eprint of the published version to anyone who needs it.]