关于n-半遗传环和n-凝聚环.pdfVIP

On -semihereditary Rings and -coherent Rings Xiaoxiang Zhang Jianlong Chen Department of Mathematics, Southeast University Nanjing 210096, P. R. China e-mail: z990303@ Abstract. Let be a ring. For fixed positive integer , is said to be left -semihereditary in case every -generated left ideal is projective. is said to be weakly -semihereditary if each -generated left (and/or right) ideal is flat. Some properties of -semihereditary rings, respectively, weakly -semihereditary rings and -coherent rings are investigated. It is also proved that is left -semihereditary if and only if it is left -coherent and weakly -semihereditary, if and only if the ring of matrices over is left 1- semihereditary if and only if the class of all -flat right -modules form the torsion-free class of a torsion theory. is left semihereditary if and only if it is left -semihereditary for all positive integers . 2000 Mathematics Subject Classification: 16D50, 16P70 Keywords: -semihereditary ring, weakly -semihereditary ring, -coherent ring There are many generalizations of hereditary rings such as, in literatures, semihered- itary rings, p.p. rings and p.f. rings. A left p.p. ring (i.e., every principal left ideal of is projective as an -module) is also called a left Rickart ring. There exists a left p.p. ring which is not right p.p. (see Chase [4] or Lam [10]). However the property that is a p.f. ring (i.e., every principal left ideal of is flat as an -module) is left-right-symmetric (see Jndrup [9] or Dauns and Fuchs [8] where