(m,n)-Hereditary Rings And (m,n)-Pure Hereditary Rings

Let m and n be two given positive integers,and R a ring.In this dissertation,we first introduce the notions of(m,n)-hereditary rings and(m,n)-semi-hereditary rings,some characterizations of these two classes of rings are given.Secondly,we in-troduce the notion of strongly(m,n)-coherent rings,it is shown that R is a strongly(m,n)-coherent ring if and only if(P m,n,Tm,n)is a hereditary cotosion pair.Then the(m,n)-projective dimension of modules and(m,n)-projective global dimension of rings are studied over strongly(m,n)-coherent rings.Finally,we introduce the no-tion of(m,n)-pure hereditary rings,the properties and equivalent characterizations of such rings are studied.