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

Abstract

Let $d_1,\dots ,d_r$ be pairwise relatively prime positive square-free integers, with $d_j\ge 2$ for all $1\le j\le r$. Using elementary matrices and combinatorial mathematics, we give a complete classification of the irreducible ${\mathbb{Z}}_{+}$-modules over the domain $\mathbb{Z}[d_1^{\frac{1}{N_1}},d_2^{\frac{1}{N_2}},\dots ,d_r^{\frac{1}{N_r}}]$, where $N_i\ge 2$ for all $1\le i\le r$. Furthermore, we explicitly construct all of these irreducible ${\mathbb{Z}}_{+}$-modules.

15 pages