Coneat submodules and coneat-flat modules
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A submodule N of a right R-module M is called coneat if for every simple right R-module S, any homomorphism N → S can be extended to a homomorphism M → S. M is called coneat-flat if the kernel of any epimorphism Y → M → 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 coneatflat 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.