Extended Reversible Rings And Abel Rings

Content: We have four parts in this paper.The first part: We introduce the grand results in the reversible ring and Abel ring and our main work in this paper.The second part: We generalize the concept of reversible rings, and pose the concepts of Extended Reversible rings, and investigate some properties about Extended Reversible rings. The following statement are the main results:Theorem 2.2.5 For a ring R the following statement are equivalent:(1) R is extended reversible ring;(2)Δ^-1R is extended reversible ring andΔbe a multiplicatively closed subset of R consisting of central regular element.Theorem 2.2.7 For a ring R the follow statement are equivalent:(1) R[ x ] is extended reversible ring;(2)Δ^-1 R[ x , x^-1] is extended reversible ring andΔbe a multiplicatively closed subset of R consisting of central regular element.Theorem 2.2.11 For an Armendariz ring R ,the following statement are equivalent:(1) R is extended reversible ring;(2) R[ x ] is extended reversible ring;Theorem 2.2.16 For an extended reversible ring R , the following statement are holded:(1) If a² = 0 for a∈R, then aR , Ra(?) N₂( R).(2) If ab = 0 for a ,b∈R, then Rab, Rba , abR , baR (?) N₂( R).(3) R is weak reversible.Theorem 2.2.17 Let R is direct sums of R₁ , R₂ ,…, R_n, then R is extended reversible ring if and only if R_i ( i = 1,2 ,…,n) is extended reversible ring.Theorem 2.2.29 If R is extended reversible ring, then it is 2-primal ring.The third part: We generalize the concept of ZI_n rings, and pose the concepts of quasi-ZI_n rings, and investigate some properties about quasi-ZI_n rings. The following statement are the main results:Theorem 3.2.3 Let A₁ , A₂,…, A_n is nonemoty subset of R , n≥2, then the following statement are equivalent:(1) R meet the property of quasi-ZI_n.(2) If A₁ A₂…A_n=0,then A_n RA_n-1 R…RA₁=0. Theorem 3.2.5 Let R is quasi-ZI_n ring, n≥3, if t is maximal in all indexes of nilpotent element of N ( R ),and n≥t,then R is 2-primal ring.The forth part: We investigate some extension properties of Abel rings and the relation between Abel rings and others rings.Theorem 4.2.2 If the trival extension of R by _RR_R is Abel ring, then R is Abel ring.Theorem 4.2.4 Let X is a set of idempotents of R , I is annihilates of X ,if R/I is Abel ring, then R is Abel ring.Theorem 4.3.2 For a Abel ring R , the following statement are holded:(1) R is stronglyÏ€-regular ring;(2) For every idempotents e∈R, eRe is stronglyÏ€-regular ring;...