Positively Graded Rings which are Unique Factorization Rings

Abstract

Let $R = \oplus _{n \in \mathbb {Z}_{0}} R_{n}$ be a positively graded ring which is a sub-ring of the strongly graded ring $S = \oplus _{n \in \mathbb {Z}} R_{n}$, where R0 is a Noetherian prime ring. It is shown that R is a unique factorization ring in the sense of (Commun. Algebra 19, 167–198, 1991) if and only if R0 is a $\mathbb {Z}_{0}$-invariant unique factorization ring and R1 is a principal (R0,R0) bi-module. We give examples of $\mathbb {Z}_{0}$-invariant unique factorization rings which are not unique factorization rings.