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CONEAT SUBMODULES AND CONEAT-FLAT MODULES

  • Buyukasik, Engin ;
  • Durgun, Yilmaz
  • Received : 2014.03.31
  • Published : 2014.11.01

Abstract

A submodule N of a right R-module M is called coneat if for every simple right R-module S, any homomorphism $N{\rightarrow}S$ can be extended to a homomorphism $M{\rightarrow}S$. M is called coneat-flat if the kernel of any epimorphism $Y{\rightarrow}M{\rightarrow}0$ 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 $M^+$ 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

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Cited by

  1. ON SOME GENERALIZATIONS OF CLOSED SUBMODULES vol.52, pp.5, 2015, https://doi.org/10.4134/BKMS.2015.52.5.1549