Classification of irreducible positive-integer modules over arbitrary positive-integer rings

Classify the irreducible \(\mathbb{Z}_+\)-modules over an arbitrary \(\mathbb{Z}_+\)-ring, beyond the special classes of rings for which explicit classifications are currently known.

Background

The paper studies irreducible $\mathbb{Z}_+$-modules over rings generated over $\mathbb{Z}$ by pairwise relatively prime, positive, $m$-th-power-free radicals. It provides an explicit classification for this particular family and establishes that every irreducible module in the family has rank $m^r$.

The authors place their result within the broader problem of determining inequivalent irreducible $\mathbb{Z}_+$-modules over finite-rank $\mathbb{Z}_+$-rings. They state that a classification for arbitrary $\mathbb{Z}_+$-rings remains unresolved, with explicit classifications available only in certain cases.