  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.
