Some Studies on Semicommutative Rings.pdf

1、In this dissertation，Semicommutative rings are mainly investigated. The whole thesis is made up of five chapters.
　　In Chapter 1，Background and basic concepts are given.
　　Letαbe an endomorphism of a ring R. In Cha

2、pter 2，we are devoted to introduce and study stronglyα-semicommutative ring，which generalizes strongly semicommuta-tive rings as well as α-rigid rings. Although the n-by-n upper triangular matrix ring Tn(R) is not strong

3、ly αˉ-semicommutative for any ring R with identity and n≥2，we show that a special subring of upper triangular matrix ring over a reduced ring is a strongly αˉ-semicommutative under some additional conditions. Moreover，it

4、 is shown that if A(R,α) is an Armendariz ring and R is α-semicommutative，then A(R,α) is stronglyα-semicommutative.
　　In Chapter 3，we introduce the concept of weakly α-semicommutative rings and investigate their prope

5、rties. We first give an example to show that semicommuta-tive rings need not be weakly α-semicommutative，and we show that R is weaklyα-semicommutative if and only if Tn(R) is weakly αˉ-semicommutative for n≥2. Also we sh

6、ow that when R is an Armendariz ring，then R is weaklyα-semicommutative if and only if the polynomial ring R[x] is weaklyα-semicommutative. Moreover，for anα-derivationδ，we introduce nil (α,δ)-semicommutative rings to inve

7、stigate the nilpotent elements in semicommutative rings.
　　An endomorphism α of a ring R is called central semicommutative if whenever for a，b ∈ R，ab = 0，then arα(b) ∈ C (R) for any r ∈ R. A ring R is called cen-tral

8、α-semicommutative if there exists a central semicommutative endomorphism. In Chapter 4，the notion of central α-semicommutative ring is introduced and studied. We show that when Δ is a multiplicatively closed subset of a

9、ring R consisting of central regular elements，then R is central α-semicommutative if and only if so isΔ-1R. It is well-known that the n by n upper triangular matrix ring Tn(R) is not central αˉ-semicommutative for any ri

10、ng R with identity when n ≥ 2. We show that some special subrings of upper triangular matrix rings over a reduced ring is centralαˉ-semicommutative under some additional conditions.
　　In Chapter 5，for a monoid M，we in

11、troduce strongly semicommutative rings relative to M . We show that every strongly M-reversible rings are strongly M-semicommutative rings，and every reduced ring is strongly M-semicommutative for any unique product monoi

12、d M . Also it is shown that for a unique product monoid M and an ideal I of R，if I is a reduced ring and R/I is strongly M-semicommutative，then R is strongly M-semicommutative. We show that some special subrings of upper