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06-01-2021 | Original Paper | Issue 3/2021

Applicable Algebra in Engineering, Communication and Computing 3/2021

On max-flat and max-cotorsion modules

Journal:
Applicable Algebra in Engineering, Communication and Computing > Issue 3/2021
Authors:
Yusuf Alagöz, Engin Büyükaşık
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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.

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