α-Symmetric Ring, Weak α-Semicommutative Ring And Weak (α,δ)-Reversible Ring

We have four parts in this paperThe first part: We introduce the grand results in the symmetric ring and semicommutative ring and reversible ring and our main work in this paper.The second part: We generalize the concept of symmetric ring, and investigate the concept ofα—Symmetric rings, and investigate some properties and extension. The following statement are the main results:Theorem 1.1.3 If R isα—Armendariz ring andα—symmetric ring, then R is symmetric ring.Theorem 1.2.3 For a ring R ,the following statement are equivalent:(1) R isα—symmetric ring;(2) eR and (1-e )R areα—symmetric rings and e is a central idempotent element which meetα( e )= e.Theorem 1.2.4 R isα—rigid ring if and only if R is reduce ring andα—symmetric ring,αis monomorphism.Theorem 1.2.7 If R is exchang integral ring,σis monomorphism, then the Nagata extension by R andσisσ—symmetric ring.The third part: We generalize the concept of semicommutative rings, and introduce the concepts of weakα—semicommutative ring, and investigate some properties and extension, and the relationship between weakα—semicommutative ring andα—Armendariz ring. The following statements are the results.Theorem 2.2.2 Let R is direct product of Ri (i∈I), then R is weakα—semicommutative ring if and only if Ri (i∈I) is weakα—semicommutative ring.Theorem 2.2.3 If R isα—rigid ring, then R is weakα—semicommutative ring.Theorem 2.2.5 R is weakα—semicommutative ring if and only if Tn ( R ) is weakα—semicommutative ring. Theorem 2.2..6 Ifα∈End ( R),which meetsα(1) = 1, then R [ x ] is weakα—semicommutative ring if and only if R[ x , x?1 ] is weakα—semicommutative ring.Theorem 2.2.7 For a reversible ring R , if R isα—semicommutative ring, then R is weakα—semicommutative ring.The forth part: We generalize the concept of reversible ring, introduce the concept of weak (α,δ)—reversible ring, and investigate the extension of weak (α,δ)—reversible ring, and the relationship between weak (α,δ)—reversible ring and weak (α,δ)—symmetric ring. The following statements are the main results: Theorem 3.2.1 Letδanα—derivation R isα—rigid ring, then is weak (α,δ)—reversible ring.Theorem 3.2.2. Let I is (α,δ)—stable ideal of R ,and weak (α,δ)—reversible. If I ? nil ( R), then R I weak (α,δ)—reversible ring if and only if R is weak (α,δ)—reversible.