Armendariz Rings,Baer Module And Extended Quasi-Baer Rings

Content: We have three parts in this paper.The first part: We introduce the grand results in the Armendariz ring and Baer ring , and our main work in the paper .The second part: We investigate some properties about Armendariz rings,deepen prehominind's results. We give more examples of Armendariz ring. The statement are the main results:Theorem 2.1.1 Let A,B are ideal of R,ii R/A is Armendariz ring ,then R/(A : B) is Armendariz rings .Theorem 2.1.4 let M is Armendariz left R—module ,then R/A(M) is Armendariz ring.Theorem 2.2.2 Let V is (A, B)- double module ,W is (B, A)-double module, then C is Armendariz ring if and only if(1) A, B is Armendariz ring.(2)V is Armendariz left A, right B- module ,W is Armendariz left B, right A-module.(3) H f(x)∈A[x],g(x)∈B[x], thenf(x)V[x]∩V[x]g(x) = 0,W[x]f(x)∩g(x)W[x]=0Theorem 2.3.1 Z- integer ring A is Armendariz ring.The third part: We pose the concept of extended quasi-Baer ring about ideal A,and Baer module, then we investigate some properties on extended . quasi -Baer Rings and Baer module. The following statement are the results:Theorem 3.1.2.2 Let f :_R M→_r N is split epimorphism. If M is Baeriquasi- Baer,p.q - Baer) module, then N is Baer (quasi - Baer, p.q - Baer) module.Theorem 3.1.2.6 let_RM is morphic module, then the following statements axe equivalent:(1) M is Baer(quasi - Baer, p.q - Baer) module.(2) N,K are submodule of M , if M/N(?)K, then M/K is Baer (quasi - Baer,p.q - Baer) module.Theorem 3.1.2.7 Let R is Abelian ring, ii R_Mi,i∈(?) = {1,2, ? ? ? , n} is Baer module, then (?)_i∈(?) M_i which is direct product of M_ii∈(?) is Baer module.Theorem 3.1.2.9 LetR is Abelian ring, if R_Mi i∈(?) = {1, 2, ? ? ? ,n} is Baer (quasi - Baer, p.q - Baer) module , then (?)_i∈(?)M_i which is direct sum of M_i, i∈(?) is Baer (quasi - Baer, p.q - Baer) module.Theorem 3.2.2.1 If R is extended quasi - Baer ring about A , then R/A is quasi - Baer ring.Theorem 3.2.2.2 Let R is reduced ring, then R is extended quasi-Baer ring about (?) if and only if R[x] is extended quasi-Baer ring about A[x] .Theorem 3.2.2.3 Leti? is reduced ring, if R is extended quasi -Baer ring about A e is semicentral idempotent in R , then ei?e is extended quasi -Baer ring about eAe .Theorem 3.2.2.4 R is extended quasi - Baer ring about A,f : R→T is homomorphism of ring, then f(R) is extended quasi-Baerring aboutTheorem 3.2.3.1 Let R_i, i∈(?) is Baer ring, then the direct sum of R_i,i∈(?) (?)_i∈(?) R_i is Baer ring.