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 PDF Compressed Postscript Version to read ISSN 0862-9544 (Printed edition) Faculty of Mathematics, Physics and Informatics Comenius University 842 48 Bratislava, Slovak Republic Telephone: + 421-2-60295111 Fax: + 421-2-65425882 e-Mail: amuc@fmph.uniba.sk Internet: www.iam.fmph.uniba.sk/amuc © 2009, ACTA MATHEMATICA UNIVERSITATIS COMENIANAE |