Positively Graded Rings which are Unique Factorization Rings

Let R=⊕n∈ℤ0Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$R = \oplus _{n \in \mathbb {Z}_{0}} R_{n}$\end{document} be a positively graded ring which is a sub-ring of the strongly graded ring S=⊕n∈ℤRn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$S = \oplus _{n \in \mathbb {Z}} R_{n}$\end{document}, 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 ℤ0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb {Z}_{0}$\end{document}-invariant unique factorization ring and R1 is a principal (R0,R0) bi-module. We give examples of ℤ0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb {Z}_{0}$\end{document}-invariant unique factorization rings which are not unique factorization rings.