MODULES WHOSE SUBMODULES ARE ESSENTIALLY EMBEDDED IN DIRECT SUMMANDS

TAKIL MUTLU F., TERCAN A.

COMMUNICATIONS IN ALGEBRA, vol.37, no.2, pp.460-469, 2009 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 37 Issue: 2
  • Publication Date: 2009
  • Doi Number: 10.1080/00927870802248688
  • Journal Name: COMMUNICATIONS IN ALGEBRA
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.460-469
  • Keywords: Cancellable module, Extending module, Finite uniform dimension, C-11-Module, C-12-Module, CS-MODULES, C-11, DCC, ACC
  • Anadolu University Affiliated: No

Abstract

A module M is said to satisfy the C-12 condition if every submodule of M is essentially embedded in a direct summand of M. It is known that the C-11 ( and hence also C-1) condition implies the C-12 condition. We show that the class of C-12-modules is closed under direct sums and also essential extensions whenever any module in the class is relative injective with respect to its essential extensions. We prove that if M is a C-12(+)-module with cancellable socle and satisfies ascending chain ( respectively, descending chain) condition on essential submodules, then M is a direct sum of a semisimple and a Noetherian ( respectively, Artinian) submodules. Moreover, a C-12-module with cancellable socle is shown to be a direct sum of a module with essential socle and a module with zero socle. An example is constructed to show that the reverse of the last result do not hold.