Commutative graded-S-coherent rings

Abstract

Recently, motivated by Anderson, Dumitrescu’s S-finiteness, D. Bennis, M. El Hajoui (2018) introduced the notion of S-coherent rings, which is the S-version of coherent rings. Let R=⊕α∈GRα\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R = \mathop \oplus \limits_{\alpha \in G} {R_\alpha }$$\end{document} be a commutative ring with unity graded by an arbitrary commutative monoid G, and S a multiplicatively closed subset of nonzero homogeneous elements of R. We define R to be graded-S-coherent ring if every finitely generated homogeneous ideal of R is S-finitely presented. The purpose of this paper is to give the graded version of several results proved in D. Bennis, M. El Hajoui (2018). We show the nontriviality of our generalization by giving an example of a graded-S-coherent ring which is not S-coherent and as a special case of our study, we give the graded version of the Chase’s characterization of S-coherent rings.