Journal of the
Korean Mathematical Society JKMS

A submodule $N$ of a right $R$-module $M$ is called \emph{coneat} if for every simple right $R$-module $S$, any homomorphism $N \to S$ can be extended to a homomorphism $M \to S$.$M$ is called \emph{coneat-flat} if the kernel of any epimorphism $Y \to M \to 0$ is coneat in $Y$. It is proven that (1) coneat submodules of any right $R$-module are coclosed if and only if $R$ is right $K$-ring; (2) every right $R$-module is coneat-flat if and only if $R$ is right $V$-ring; (3) coneat submodules of right injective modules are exactly the modules which have no maximal submodules if and only if $R$ is right small ring. If $R$ is commutative, then a module $M$ is coneat-flat if and only if $M^+$ is $m$-injective. Every maximal left ideal of $R$ is finitely generated if and only if every absolutely pure left $R$-module is $m$-injective. A commutative ring $R$ is perfect if and only if every coneat-flat module is projective. We also study the rings over which coneat-flat and flat modules coincide.

Keywords: neat submodule, coclosed submodule, coneat submodule, coneat-flat module, absolutely neat module

MSC numbers: 16D10, 16D40, 16D80, 16E30