CONEAT SUBMODULES AND CONEAT-FLAT MODULES

Title & Authors
CONEAT SUBMODULES AND CONEAT-FLAT MODULES

Abstract
A submodule N of a right R-module M is called coneat if for every simple right R-module S, any homomorphism $\small{N{\rightarrow}S}$ can be extended to a homomorphism $\small{M{\rightarrow}S}$. M is called coneat-flat if the kernel of any epimorphism $\small{Y{\rightarrow}M{\rightarrow}0}$ 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 $\small{M^+}$ 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;
Language
English
Cited by
1.
ON SOME GENERALIZATIONS OF CLOSED SUBMODULES,;

대한수학회보, 2015. vol.52. 5, pp.1549-1557
1.
ON SOME GENERALIZATIONS OF CLOSED SUBMODULES, Bulletin of the Korean Mathematical Society, 2015, 52, 5, 1549
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