ABELIAN MODULES

Agayev N., Gungoroglu G., Harmanci A., Halicioglu S.

ACTA MATHEMATICA UNIVERSITATIS COMENIANAE, vol.78, no.2, pp.235-244, 2009 (ESCI) identifier identifier

  • Publication Type: Article / Article
  • Volume: 78 Issue: 2
  • Publication Date: 2009
  • Journal Name: ACTA MATHEMATICA UNIVERSITATIS COMENIANAE
  • Journal Indexes: Emerging Sources Citation Index (ESCI), Scopus
  • Page Numbers: pp.235-244
  • Hacettepe University Affiliated: Yes

Abstract

In this note, we introduce abelian modules as a generalization of abelian rings. Let R be an arbitrary ring with identity. A module M is called abelian if, for any m is an element of M and any a is an element of R, any idempotent e is an element of R, mae = mea. We prove that every reduced module, every symmetric module, every semicommutative module and every Armendariz module is abelian. For an abelian ring R, we show that the module M-R is abelian iff M[x](R[x]) is abelian. We produce an example to show that M[x, alpha] need not be abelian for an abelian module M and an endomorphism alpha of the ring R. We also prove that if the module M is abelian, then M is p.p.-module iff M[x] is p.p.-module, M is Baer module iff M[x] is Baer module, M is p.q.-Baer module iff M[x] is p.q.-Baer module.