On W-ideals & Annihilators Of Rings And Primary Decomposition Of Modules

In this paper, We discuss three aspects of ring theory:First, we discuss the relation between one sided ff-ideals of R and one~sided annihilator ideals of Mn(R).The main conclusions are as following:Let R be a ring and Mn(R), (nl), be matrix rings, S be any subset of Mn(R),then:1: As right R-module, if the left W-change of S is generated by two elements, then every ru (S)(S)is a middle term of a short exact sequence where the first and the last non-zero terms are both finite direct æ­¶m of right ff-ideals of R;2: As right R-module, if the left ff-change of S is generated by K elemants and k^3, then every ru (g)(S) ig tne middle term of a short exact sequencewhere the first non-zero term is a finite direct m of some annihilators of some subsets of Mn(R) whose left W-change is generated by k-1 elements and the last non-zero term is a finite direct æ­¶m of some right ff-ideals;3: By l and 2, we have: 1) :R is right Il-coherentif and only if every right annihilator ideal and its right W-ideals are finitely generated;2):Every integral domains is right and leftn-coherent.Secondly, we discuss the relation of annihilator ideals of any subset of R and of Mn(R) :1: Let R be a ring and AC K be any subset ofR, as a right R-module, r(A) is isomorphic to adirect summand of a right annihilator of some subset of Mn(R) ;2: Let R be a ring, AcR be a finite subset of R, then, as right R-module, r(A) is isomorphic to a direct summand of ru (R)(B) for some finite subset3: By l and 2, we know if R is n-coherent, then ali annihilator ideals are finitely generated; If R is coherent, then ali annihilators any finite subset of R are finitely generated.Thirdly, we discuss two uniqueness theorera.3 of primary decomposition of modules over commutative rings, without the condition that the modules are finitely generated.