Moroccan Journal of Algebra and Geometry with Applications

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On rings whose finitely generated flat submodules of free
modules are finitely presented

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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