¦³-n-coherent Rings

Coherent ring is one of the most important and intriguing aspect of ring theory. The main theme of the paper was to study n-coherence relative to an arbitrary (hereditary) torsion theories, and to obtain characterizations of left r-n-coherent rings for a fixed positive integer n. We gave various equivalent conditions for a ring to be r-n-coherent. Meanwhile we discussed the some properties of r-n-coherent. It was shown that there are many similarities between coherent rings and Ï„-n-coherent rings, then some known results on coherent rings were obtained as corollaries. The paper consists of three chapters, and in chapter one we introduced the revelant background of coherence and the main directions in coherence researching.In chapter two, we not only defined r-n-presented modules , but also proved that for a fixed positive integer n and Ï„-(n -1)-presented module M, then the followings are equivalent.(1) M is Ï„-n-presented.(2) ExtRn-1(M,limNi = limExtRn-1(M,Ni), for each direct system {Ni} of torsion free R-module.(3) TorRn-1 (N+, M) = ExtRn-1(M, N)+, for every torsion free module N.(4) TorRn-1(E+, M) = 0, for each torsion free injective module E.Then we also denned r-n-coherent modules, r-n-absolutely pure modules and Ï„-n-falt modules which were based on r-n-presented modules,and we investigated the relationship between r-n-absolutely pure modules and r-n-falt modules. At last we characterized it in terms of relative n-absolutely pure modules and relative n-flat modules.In chapter three we gave definitions of w-dule modules as a generalized dule module and r-absolutely pure resolutions. We investigated their property on Ï„-n-coherent rings, then we got that provied that R is a commutative coherent ring and rR is torsionless r-absolutely pure R-module, then any r-n-presented moduleis reflexive module. Lastly, we concerned about whether the r-n-coherence can be kept between two rings changed by a certain morphism.