Absolutely s-Pure Modules and Neat-Flat Modules

Abstract

Let R be a ring with an identity element. We prove that R is right Kasch if and only if injective hull of every simple right R-modules is neat-flat if and only if every absolutely pure right R-module is neat-flat. A commutative ring R is hereditary and noetherian if and only if every absolutely s-pure R-module is injective and R is nonsingular. If every simple right R-module is finitely presented, then (1) _R R is absolutely s-pure if and only if R is right Kasch and (2) R is a right -CS ring if and only if every pure injective neat-flat right R-module is projective if and only if every absolutely s-pure left R-module is injective and R is right perfect. We also study enveloping and covering properties of absolutely s-pure and neat-flat modules. The rings over which every simple module has an injective cover are characterized.

Key Words:

• Absolutely s-pure module
• Injective cover
• Kasch ring
• Neat submodule
• Simple-projective module
• s-Pure submodule

2010 Mathematics Subject Classification:

• 16D10
• 16D40
• 16D60
• 16E30
• 16L60

ACKNOWLEDGMENTS

Some part of this paper was written while the second author was visiting Padova University, Italy. He wishes to thank the members of the Department of Mathematics for their kind hospitality.

Notes

Communicated by E. Puczylowski.