Vol. LXXVIII, 2 (2009)
p. 235 - 244

Abelian modules

N. Agayev, G. Güngöroğlu, A. Harmanci and S. Halıcıoğlu

Received: April 09, 2008;   Revised: June 20, 2008;   Accepted: June 26, 2008

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 Î M and any a Î R, any idempotent e Î 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.

Keywords:  semicommutative modules; Armendariz modules; abelian modules; reduced modules; symmetric modules.  

AMS Subject classification: Primary:  16U80  

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Acta Mathematica Universitatis Comenianae
ISSN 0862-9544   (Printed edition)

Faculty of Mathematics, Physics and Informatics
Comenius University
842 48 Bratislava, Slovak Republic  

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