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

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