Slightly (m, n)-coherent rings and (m, n)-homological dimensions

Abstract

Let R be a ring and m, n two fixed positive integers. In this article, R is defined to be left slightly (m, n)-coherent if each n-generated submodule of the left R-module R^m is (m, n)-projective. It is shown that R is left slightly (m, n)-coherent if and only if the cotorsion pair $({\mathcal{P}}_{m,n},{\mathcal{I}}_{m,n})$ is hereditary if and only if R is left (m, n)-coherent and the cotorsion pair $({\mathcal{F}}_{m,n},{\mathcal{C}}_{m,n})$ is hereditary, where ${\mathcal{P}}_{m,n}$ (resp., ${\mathcal{I}}_{m,n}$) stands for the class of all (m, n)-projective (resp., (m, n)-injective) left R-modules, and ${\mathcal{F}}_{m,n}$ (resp., ${\mathcal{C}}_{m,n})$ denotes the classes of all (m, n)-flat (resp., (m, n)-cotorsion) right R-modules. As applications, (m, n)-homological dimensions of modules and rings over slightly (m, n)-coherent rings are studied.

Keywords:

• Slightly (m, n)-coherent ring; (m, n)-injective module; (m, n)-projective module; (m, n)-flat module; (m, n)-homological dimension

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 16D40
• 16D50
• 16P70

Acknowledgement

The authors are grateful to the referee for the careful checking of this article and some helpful comments.

Additional information

Funding

This work was partially supported by the National Natural Science Foundation of China (11861055, 11761060).