THE EXTENDING CONDITION RELATIVE TO SETS OF SUBMODULES


Birkenmeier G. F. , TERCAN A. , CELEP YÜCEL C.

COMMUNICATIONS IN ALGEBRA, cilt.42, sa.2, ss.764-778, 2014 (SCI İndekslerine Giren Dergi) identifier identifier

  • Cilt numarası: 42 Konu: 2
  • Basım Tarihi: 2014
  • Doi Numarası: 10.1080/00927872.2012.723084
  • Dergi Adı: COMMUNICATIONS IN ALGEBRA
  • Sayfa Sayıları: ss.764-778

Özet

A module M is called an extending (or CS) module provided that every submodule of M is essential in a direct summand of M. We call a module ?-extending if every member of the set ? is essential in a direct summand where ? is a subset of the set of all submodules of M. Our focus is the behavior of the ?-extending modules with respect to direct sums and direct summands. By obtaining various well-known results on extending modules and generalizations as corollaries of our results, we show that the ?-extending concept provides a unifying framework for many generalizations of the extending notion. Moreover, by applying our results to various sets ?, including the projection invariant submodules, the projective submodules, and torsion or torsion-free submodules of a module, we obtain new results including a characterization of the projection invariant extending Abelian groups.