Some Studies On Extended Commutative Ring

In this paper, we introduce some studies on extended commutative ring,investigate various generalization of reversible rings and reflexive rings, which based on the results given by former papers. Some rings are introduced and investigated, for example, the skew strongly M-reversible ring, strongly ?-reflexive ring, strongly reflexive ring, strongly M-reflexive ring and strongly M-?-reflexive ring and so on. As a result, some well-known concepts and results are generalized.This paper consists of five chapters as follows.In Chapter 1, some backgrounds and preliminaries are given. We introduced the concepts of reversible ring, reflexive ring, reduced ring, rigid ring and so on. Some results of these rings are given.In Chapter 2, the notion of skew strongly M-reversible ring is introduced as a generalization of strongly M-reversible ring. We study the examples and properties of skew strongly M-reversible ring. For a ring R and a monoid M with a monoid homomorphism ?:M?End(R), we prove that if R is M-compatible and a?R is a idempotent, then R is skew strongly M-reversible if and only if aR and(1-a)R are skew strongly M-reversible. It is shown that if G is a finitely generated Abelian group,then G is torsion-free if and only if there exists a ring R with |R|?2 such that R is skew strongly G-reversible. Moreover, we prove that if R is a right Ore ring with classical right quotient ring Q, then R is skew strongly M-reversible if and only if Q is skew strongly M-reversible.In Chapter 3, we introduce the concept of strongly reflexive ring and strongly M-reflexive ring. Some characterizations and various extensions of the two classes of rings are obtained. It is proved that R is strongly reflexive if and only if R[x] is strongly reflexive if and only if R[x; x-1] is strongly reflexive. For a right Ore ring R with classical right quotient ring Q, we show that R is strongly reflexive if and only if Q is strongly reflexive. Moreover, we prove that if M is a u.p.-monoid and R is a reduced ring, then R is strongly M-reflexive.In Chapter 4, we consider the ?-reflexivity over which polynomial rings are?-reflexive and call them strongly ?-reflexive rings. A number of properties of this version are established. For an endomorphism ? of R, we prove that R[x] is strongly?-reflexive if and only if R[x; x-1] is strongly ?-reflexive. For an Armendariz ring R, R is ?-reflexive if only if R is strongly ?-reflexive if and only if R[x; x-1] is strongly?-reflexive. We also show that for a right Ore ring R with Q its classical right quotient ring, if R[x] is ?-rigid, then R is strongly ?-reflexive if and only if Q is strongly?-reflexive.In Chapter 5, the notion of strongly M-?-reflexive ring is introduced as a generalization of strongly ?-reflexive ring and strongly M-reflexive ring. Some properties and examples of strongly M-?-reflexive ring are given. It is shown that if R is a right Ore ring such that R[M] is ?-rigid, then R is strongly M-?-reflexive if and only if its classical right quotient ring Q is strongly M-?-reflexive. Also, we prove that if R is a reduced strongly M-?-reflexive ring, then R is strongly(M×N)-?-reflexive,where M and N are two u.p.-monoids.