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.
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.