2017/09/28 by Daniel Bravo, Bravo, Daniel, Sergio Estrada +3
Mathematics · Medicine · #18G25 #18G35 #Algebraic structures and combinatorial models #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Neurosurgical Procedures and Complications #Rings and Algebras (math.RA) #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.1709.10160
openalex publication_date 2017/09/28 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
We prove that, for any n \≥ 2, the classes of rmFPn-injective\nmodules and of rmFPn-flat modules are both covering and preenveloping\nover any ring R. This includes the case of rmFP\∞-injective and\n rmFP\∞-flat modules (i.e. absolutely clean and, respectively, level\nmodules). Then we consider a generalization of the class of (strongly)\nGorenstein flat modules - the (strongly) Gorenstein AC-flat modules (cycles of\nexact complexes of flat modules that remain exact when tensored with any\nabsolutely clean module). We prove that some of the properties of Gorenstein\nflat modules extend to the class of Gorenstein AC-flat modules; for example we\nshow that this class is precovering over any ring R. We also show that (as in\nthe case of Gorenstein flat modules) every Gorenstein AC-flat module is a\ndirect summand of a strongly Gorenstein AC-flat module. When R is such that\nthe class of Gorenstein AC-flat modules is closed under extensions, the\nconverse is also true. We also prove that if the class of Gorenstein AC-flat\nmodules is closed under extensions, then this class of modules is covering.\n