On rings whose finitely generated flat submodules of free
modules are finitely presented
Samir Bouchiba , Mouhssine El-Arabi & Youssef Najem
Department of Mathematics, Faculty of sciences, Moulay Ismail university of Meknes, Meknes, Morocco
Pages 122-131 | Received 15 October 2021, Accepted 02 February 2022, Published 01 June 2022
Abstract
A ring R is right coherent if any finitely generated submodule of a free right module is finitely presented. The main goal of this paper is to study the rings on which any finitely generated flat submodule of a free right module is finitely presented, and thus projective. For this sake, we investigate homological properties of specific classes of modules, namely, the \mathcal F_1-flat modules and the \mathcal F_1^{\mathrm{fp}}-flat modules, where \mathcal F_1 stands for the class of right modules of flat dimension at most one and \mathcal F_1^{\mathrm{fp}} its subclass consisting of finitely presented elements. The introduced class of rings that we term \mathcal F_1^{\mathrm{fp}}-coherent rings turns out to behave nicely with respect to \mathcal F_1^{\mathrm{fp}}-flat modules as do coherent rings with respect to flat modules. In fact, we prove that a ring R is \mathcal F_1^{\mathrm{fp}}-coherent if and only if any direct product of \mathcal F_1^{\mathrm{fp}}-flat modules is \mathcal F_1^{\mathrm{fp}}-flat. Also, we show that the class of \mathcal F_1^{\mathrm{fp}}-coherent rings includes coherent rings, perfect rings, semi-hereditary rings and integral domains.
Keywords: \mathcal F_1-flat module, \mathcal F_1^{\mathrm{fp}}-flat module, \mathcal F_1^{\mathrm{fp}}-coherent ring, finitistic flat dimension, little finitistic dimension.
MSC numbers: Primary 13D02, 13D05, 13D07; Secondary 16E05, 16E10.
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