Some Properties of Rickart Modules

Abstract

R birimli bir halka, M saº g R-mod¸l ve M nin endomorÖzma halkas¨ S = EndR(M) olsun. Her f 2 S iÁin rM(f) = eM olacak biÁimde e2 = e 2 S varsa (denk olarakKerf,Mmod¸l¸n¸nbirdirekttoplanan¨ise)MyeRickartmod¸lad¨verilmi?stir[8]. BuÁal¨?smadaRickartmod¸llerinˆzellikleriincelenmeyedevamedilmi?stir. M birRickart mod¸l olmak ¸zere, M nin S-kat¨ (s¨ras¨yla S-indirgenmi?s, S-simetrik, S-yar¨ deºgi?smeli, S-Armendariz)mod¸l olmas¨ iÁin gerek ve yeter ?sart¨n S nin kat¨ (s¨ras¨yla indirgenmi?s, simetrik, yar¨ deºgi?smeli, Armendariz) halka olduºgu gˆsterilmi?stir. M[x], S[x] halkas¨na gˆre Rickart mod¸l iken M nin de Rickart mod¸l oldugu,tersinin M nin S-Armendariz olmas¨ durumunda doºgru olduºgu ispatlanm¨?st¨r. Ayrıca bir M mod¸l¸n¸n Rickart ol- mas¨iÁingerekveyeter?sart¨nhersaºgmod¸l¸nM-temelprojektifolduºgueldeedilmi?stir.Let Rbeanarbitraryringwithidentity and M aright R-module with S =EndR(M). Following [8],the module M is called Rickart if for any f 2 S, rM(f) = eM for some e2 = e 2 S, equivalently, Kerf is a direct summandofM. Inthispaper,wecontinuetoinvestigatepropertiesofRickart modules. For a Rickart module M, we prove that M is S-rigid (resp., S- reduced, S-symmetric, S-semicommutative, S-Armendariz) if and only if its endomorphism ring S is rigid (resp., reduced, symmetric, semicommutative, Armendariz). We also prove that if M[x]is a Rickart module with respect to S[x], then M is Rickart, the converse holds if M is S-Armendariz. Among others it is also shown that M is a Rickart module if and only if every right R-module is M-principally projective.