Neat-Flat Modules

TL;DR

This paper characterizes neat-flat modules over rings, establishing their equivalence with simple-projective modules and exploring conditions under which they are projective, especially over rings with specific properties like right CS and C-rings.

Contribution

It provides new characterizations of neat-flat modules, linking them to simple-projective modules and identifying conditions for projectivity over certain classes of rings.

Findings

Neat-flat modules are equivalent to simple-projective modules.

Neat-flat modules are projective iff the ring is right The paper characterizes neat-flat modules over rings, linking them to projectivity and specific ring properties.

Abstract

Let $R$ be a ring and $M$ be a right $R$-module. $M$ is called neat-flat if any short exact sequence of the form $0\to K\to N\to M\to 0$ is neat-exact i.e. any homomorphism from a simple right $R$-module $S$ to $M$ can be lifted to $N$. We prove that, a module is neat-flat if and only if it is simple-projective. Neat-flat right $R$-modules are projective if and only if $R$ is a right $\sum$-$CS$ ring. Every finitely generated neat-flat right $R$-module is projective if and only if $R$ is a right $C$-ring and every finitely generated free right $R$-module is extending. Every cyclic neat-flat right $R$-module is projective if and only if $R$ is right $CS$ and right $C$-ring. Some characterizations of neat-flat modules are obtained over the rings whose simple right $R$-modules are finitely presented.