STRONGLY GRADED MODULES AND POSITIVELY GRADED MODULES WHICH ARE UNIQUE FACTORIZATION MODULES

Let M = ⊕n∈Z Mn 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∈Z0 Mn to be a UFM over positively graded domain R = ⊕n∈Z0 Dn.