Right Reversible Rings And Weak α-symmetric Rings

We have three parts in this paper.The first part:We introduce the grand results in the right reversible rings and Weakα-symmetric rings,and our main work in the paper.The second part:We generalize the cencept of Right reversible rings,and pose the concept of reversible rings,and investigate some properties about Right reversible rings.The following statement are the main result:Theorem1.2.3 Let R is right reversible rings then for any n ,the n×n upper triangular mateix ring Tn ( R) is right reversible rings.Theorem1.2.4 If R is right reversible ring,then T ( R, R) is right reversible ring.Theorem1.2.6 Suppose R is symmetric,if R I is right reversible ring, I is ideal and annihilators of R ,then R is right reversible rings.Theorem 1.2.7 For a ring R :(i)Suppose R is Armendriz rings, R [x ] is right reversible,then R is right reversible.(ii)If R [x ] is right reversible rings ,then R[x,x^-1] is right reversible rings.The third part:we pose the concept of weakly symmetric ring,and investigate some properties on weakα-symmetric ring.Theorem 2.2.3 Letαbe a monomorphism of a ring R ,then weakα-symmetric ring of reduced ring isα-rigid ring.Theorem2.2.6 Let R is weakα-symmetric ring,then for any n, then T_n ( R) is weakα-symmetric ring.Theorem2.3.2 Suppose 1 is the idention of A, B ,then the following statements are equivalent:(i)There are epimorphismsα∈EndT ,such that T is weakα-symmetric ring;(ii)There are epimorphismsβ∈EndA and epimorphismsγ∈EndB ,such that A is weakβ-symmetric ring, B is weakγ-symmetric ring ,and V =0,W =0.Theorem 2.3.3 (i)If ring R is finit direct product of weakα-symmetric ring,then R is weakα-symmetric ring.(ii)If ring R is finit sub direct product of weakα-symmetric ring,then R is weakα-symmetric ring.