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1、http://www.paper.edu.cnOnn-semihereditaryRingsandn-coherentRingsXiaoxiangZhangJianlongChenDepartmentofMathematics,SoutheastUniversityNanjing210096,P.R.Chinae-mail:z990303@seu.edu.cnAbstract.LetRbearing.Forfixedpositiveintegern,Rissaidtobeleftn-semihereditaryinc
2、aseeveryn-generatedleftidealisprojective.Rissaidtobeweaklyn-semihereditaryifeachn-generatedleft(and/orright)idealisflat.Somepropertiesofn-semihereditaryrings,respectively,weaklyn-semihereditaryringsandn-coherentringsareinvestigated.ItisalsoprovedthatRisleftn-se
3、mihereditaryifandonlyifitisleftn-coherentandweaklyn-semihereditary,ifandonlyiftheringofn£nmatricesoverRisleft1-semihereditaryifandonlyiftheclassofalln-flatrightR-modulesformthetorsion-freeclassofatorsiontheory.Risleftsemihereditaryifandonlyifitisleftn-semihered
4、itaryforallpositiveintegersn.2000MathematicsSubjectClassification:16D50,16P70Keywords:n-semihereditaryring,weaklyn-semihereditaryring,n-coherentringTherearemanygeneralizationsofhereditaryringssuchas,inliteratures,semihered-itaryrings,p.p.ringsandp.f.rings.Aleft
5、p.p.ringR(i.e.,everyprincipalleftidealofRisprojectiveasanR-module)isalsocalledaleftRickartring.Thereexistsaleftp.p.ringwhichisnotrightp.p.(seeChase[4]orLam[10]).HoweverthepropertythatRisap.f.ring(i.e.,everyprincipalleftidealofRisflatasanR-module)isleft-right-sy
6、mmetric(seeJÁndrup[9]orDaunsandFuchs[8]whereap.f.ringissaidtobetorsion-free).Indeed,foraringRandafixedpositiveintegern,JÁndrup[9]hasprovedthatifalln-generatedleftidealsareflat,thenalln-generatedrightidealsareflat,too.SucharingRwasalsodiscussedbyShamsuddin[11]unde
7、rtheterminologyn-semihereditary.Butweshallcallitaweaklyn-semihereditaryringsinceZhuandTandefinealeft(resp.right)n-semihereditaryringtobearingRwhosen-generatedleft(resp.right)idealsareallprojective.Thus“leftp.p.”=“leftRickart”=“left1-semihereditary”and“p.f.”=“to
8、rsion-free”=“weakly1-semihereditary”.FollowingShamsuddin[11],aringRissaidtobeleftn-coherentifeveryn-generatedleftidealofRisfinitelypresented.Notethatthisdefinitionisatoddswithanother
