Strongly Graded Modules and Positively Graded Modules which are Unique Factorization Modules

Year 2024, Volume: 36 Issue: 36, 1 - 15, 12.07.2024

Iwan Ernanto Indah E. Wijayanti Akira Ueda

https://doi.org/10.24330/ieja.1404435

https://izlik.org/JA64JU94BR

Abstract

Let $M=\oplus_{n\in \mathbb{Z}}M_{n}$ be a strongly graded module over strongly graded ring $D=\oplus_{n\in \mathbb{Z}} D_{n}$. In this paper, we
prove that if $M_{0}$ is a unique factorization module (UFM for short) over $D_{0}$ and $D$ is a unique factorization domain (UFD for short), then $M$ is a UFM over $D$. Furthermore, if $D_{0}$ is a Noetherian domain, we give a necessary and sufficient condition for a positively graded module $L=\oplus_{n\in \mathbb{Z}_{0}}M_{n}$ to be a UFM over positively graded domain $R=\oplus_{n\in \mathbb{Z}_{0}}D_{n}$.

Keywords

Graded ring , graded module , unique factorization module

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