Some Studies On Endomorphisms Of Rings

In this paper, we first introduce the notion of left α-semicommutative rings and α-sc rings, which based on the the research of semicommutative rings. We next consider some basic properties of left α-semicommutative rings and α-sc rings, including some examples needed in the process. Moreover, we mainly sduty some extensions of left α-semicommutative rings and α-sc rings. In particular, we introduce the notion ofα-γf rings to extend the concept of reflexive rings. We first consider properties and basic extensions of α-γf rings. Next we investigate the α-γfty of some kinds of polynomial rings and classical right quotient rings.We have four parts in this paper:Chaper1:As preliminaries, we introduce the notion of semicommutative rings、reversible rings、symmetric rings and reflexive rings. In addition, we summarize the research overviews of semicommutative rings and reflexive rings. At last, we provide this paper’s research ideas and conclusions.Chaper2:We introduce the notion of left α-semicommutative rings by observing the right α-semicommutative rings, and we also discuss the left α-semicommutativity of matrix rings and some kinds of polynomial rings.Chaper3:We study rings endomorphisms with the semicommutive condition. The notion of a-sc rings, a generalization of semicommutative rings, is introduced. Moreover, we discuss the relationships between α-sc rings and α-shifting rings, and we study the extensions of α-sc rings, including trivial extensions and classical quotient rings. Consequently, we prove that polynomial ring R[x]、Laurent polinomial ring R[x;x-1]、skew polynomial ring R[x;α] of R are some necessary and sufficient conditions of α-sc rings.Chaper4:We study rings endomorphisms with the reflexive condition. The notion of α-rf rings, a generalization of reflexive rings, is introduced, the relations between α-γf rings and related rings are discussed. We next study the extensions ofα-γf rings, including trivial extensions and Dorroh extensions. As a consequense, we consider the α-γfty of some kinds of polynomial rings and classical right quotient rings.