@Article{JMS-54-4,
author = {Zhichao, Chen and Jiayi, Cai and Meng, Lingchao and Libin, Li},
title = {Non-Negative Integer Matrix Representations of a $\mathbb{Z}_{+}$-Ring},
journal = {Journal of Mathematical Study},
year = {2021},
volume = {54},
number = {4},
pages = {357--370},
abstract = {<p style="text-align: justify;">The $\mathbb{Z}_{+}$-ring is an important invariant in the theory of tensor category. In this paper, by using matrix method, we describe all irreducible $\mathbb{Z}_{+}$-modules over a $\mathbb{Z}_{+}$-ring $\mathcal{A}$, where $\mathcal{A}$ is a commutative ring with a $\mathbb{Z}_{+}$-basis{$1$, $x$, $y$, $xy$} and relations: $$ x^{2}=1,\;\;\;\;\; y^{2}=1+x+xy.$$We prove that when the rank of $\mathbb{Z}_{+}$-module $n\geq5$, there does not exist irreducible $\mathbb{Z}_{+}$-modules and when the rank $n\leq4$, there exists finite inequivalent irreducible $\mathbb{Z}_{+}$-modules, the number of which is respectively 1, 3, 3, 2 when the rank runs from 1 to 4.</p>},
issn = {2617-8702},
doi = {https://doi.org/10.4208/jms.v54n4.21.02},
url = {https://global-sci.com/article/87680/non-negative-integer-matrix-representations-of-a-mathbbz-ring}
}