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

Let M=⊕n∈ ZMₙ be a strongly graded module over strongly graded ring D=⊕n∈ Z Dₙ. In this paper, we prove that if M₀ is a unique factorization module (UFM for short) over D₀ and D is a unique factorization domain (UFD for short), then M is a UFM over D. Furthermore, if D₀ is a Noetherian domain, we give a necessary and sufficient condition for a positively graded module L=⊕_n∈ Z₀Mₙ to be a UFM over positively graded domain R=⊕_n∈ Z₀Dₙ.