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

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 $R=\oplus_{n\in \mathbb{Z}_{0}}D_{n}$.