Irreducible $\mathbb{Z}_+$-modules over some $\mathbb{Z}_+$-rings

Let $d_1,\ldots,d_r$ be pairwise relatively prime positive square-free integers, with $d_j\geq2$ for all $1\leq j\leq r$. Using elementary matrices and combinatorial mathematics, we give a complete classification of the irreducible $\mathbb{Z}_+$-modules over the domain $\mathbb{Z}[d^{\frac{1}{N_1}}