On max-flat and max-cotorsion modules

Abstract

In this paper, we continue to study and investigate the homological objects related to s-pure and neat exact sequences of modules and module homomorphisms. A right module A is called max-flat if $${\text {Tor}}_{1}^{R}(A, R/I)= 0$$ for any maximal left ideal I of R. A right module B is said to be max-cotorsion if $${\text {Ext}}^{1}_{R}(A,B)=0$$ for any max-flat right module A. We characterize some classes of rings such as perfect rings, max-injective rings, SF rings and max-hereditary rings by max-flat and max-cotorsion modules. We prove that every right module has a max-flat cover and max-cotorsion envelope. We show that a left perfect right max-injective ring R is QF if and only if maximal right ideals of R are finitely generated. The max-flat dimensions of modules and rings are studied in terms of right derived functors of $$-\otimes -$$ . Finally, we study the modules that are injective and flat relative to s-pure exact sequences.