FP_n-injective and FP_n-flat covers and preenvelopes and Gorenstein AC-flat covers

Daniel Bravo, Sergio Estrada, Alina Iacob

Research output: Contribution to journalArticlepeer-review

14 Scopus citations

Abstract

We prove that, for any n ≥ 2, the classes of FPn-injective modules and of FPn-flat modules are both covering and preenveloping over any ring R. This includes the case of FP-injective and FP-flat modules (i.e., absolutely clean and, respectively, level modules). Then we consider a generalization of the class of (strongly) Gorenstein flat modules, i.e., the (strongly) Gorenstein AC-flat modules (cycles of exact complexes of flat modules that remain exact when tensored with any absolutely clean module). We prove that some of the properties of Gorenstein flat modules extend to the class of Gorenstein AC-flat modules; for example, we show that this class is precovering over any ring R. We also show that (as in the case of Gorenstein flat modules) every Gorenstein AC-flat module is a direct summand of a strongly Gorenstein AC-flat module. When R is such that the class of Gorenstein AC-flat modules is closed under extensions, the converse is also true. Moreover, we prove that if the class of Gorenstein AC-flat modules is closed under extensions, then it is covering.

Original languageAmerican English
JournalAlgebra Colloquium
Volume25
StatePublished - 2018

Disciplines

  • Physical Sciences and Mathematics

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