Irreducible ${\mathbb{Z}}_{+}$-modules over $\mathbb{Z}[\sqrt[m]{d_1},\sqrt[m]{d_2},\dots ,\sqrt[m]{d_r}]$

Abstract

Let $d_1,d_2,\dots ,d_r\ge 2$ be $m$-th power free positive integers. In this paper, we provide explicit representations of all irreducible ${\mathbb{Z}}_{+}$-modules over $\mathbb{Z}[\sqrt[m]{d_1},\sqrt[m]{d_2},\dots ,\sqrt[m]{d_r}]$, where ${\mathbb{Z}}_{+}$ denotes the semiring of non-negative integers.