Neat-flat Modules

Abstract

Let R be a ring. A right R-module M is said to be neat-flat if the kernel of any epimorphism Y → M is neat in Y, i.e., the induced map Hom(S, Y) → Hom(S, M) is surjective for any simple right R-module S. Neat-flat right R-modules are projective if and only if R is a right -CS ring. Every cyclic neat-flat right R-module is projective if and only if R is right CS and right C-ring. It is shown that, over a commutative Noetherian ring R, (1) every neat-flat module is flat if and only if every absolutely coneat module is injective if and only if R ≅ A × B, wherein A is a QF-ring and B is hereditary, and (2) every neat-flat module is absolutely coneat if and only if every absolutely coneat module is neat-flat if and only if R ≅ A × B, wherein A is a QF-ring and B is Artinian with J ²(B) = 0.

Key Words:

• Closed submodule
• (Co)neat submodule
• Extending module
• Neat-flat module
• QF-ring

2010 Mathematics Subject Classification:

• 16D10
• 16D40
• 16E30

ACKNOWLEDGMENTS

Some part of this article was written while the second author was visiting Padova University, Italy. He would like to thank the members of the Department of Mathematics of Padova University for their hospitality. The authors are grateful to the referee for carefully reading the paper and valuable comments that improved presentation of the paper.

Notes

Communicated by E. Puczylowski.