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

Authors : Iwan Ernanto, Indah E Wijayanti, Akira Ueda

Pages : 1-15

Doi:10.24330/ieja.1404435

View : 106 | Download : 117

Publication Date : 2024-07-12

Article Type : Research Paper

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