Abelian modules

Agayev N., Güngöroǧlu G., Harmanci A., HALICIOĞLU S.

Acta Mathematica Universitatis Comenianae, vol.78, no.2, pp.235-244, 2009 (Scopus) identifier identifier

  • Publication Type: Article / Article
  • Volume: 78 Issue: 2
  • Publication Date: 2009
  • Journal Name: Acta Mathematica Universitatis Comenianae
  • Journal Indexes: Scopus
  • Page Numbers: pp.235-244
  • Keywords: semicommutative modules, Armendariz modules, abelian modules, reduced modules, symmetric modules
  • Ankara 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 ∈ M and any a ∈ R, any idempotent e ∈ 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 MR is abelian iff M[x]R[x] is abelian. We produce an example to show that M[x, α] need not be abelian for an abelian module M and an endomorphism α 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.