Nilpotent Elements And Special Ring Extensions

Commutative rings form a very special subclass of rings, which shows quite differ-ent behaviour from the general case. More generally, Cohn introduced the notion of a reversible ring [14] to study commutativity of a non-commutative ring, and proved that a ring R is reduced if and only if it is semiprime and reversible. In recent years, various generalizations and results related to reversible rings such as Armendariz rings, McCoy rings, Baer rings and reflexive rings are obtained. It is well-known that the Kothe Conjecture has close connection with the nilpotent elements of a ring. The Kothe Conjecture states that if a ring R has no nonzero nil ideals then R has no nonzero nil one-sided ideals. Although for more than70years significant progress has been made, it is still open in general. It was shown in [14] that the Kothe Conjecture holds for the class of reversible rings. Inspired by this fact, we investigate various generalizations of reversible rings and related rings in this dissertation. Some new rings are introduced and investigated. As a result, some well-known concepts and results are generalized.This paper consists of six chapters.In Chapter1, some backgrounds and preliminaries are given.In Chapter2, the notion of right generalized p.q.-Baer modules (or simply right GPQ modules) is introduced as a non-trivial generalization of the notion of right p.q.-Baer modules. We study on the relationship between the GPQ property of a module MR and various quasi-Armendariz properties. We prove that if MR is a right GPQ module, then MR is a quasi-Armendariz module. For a right module MR, it is shown that MR is a right GPQ module if and only if M[x]R[x] is a right GPQ module. Some properties and examples are given and some known results are extended. We also consider the weak Armendariz property of the formal triangular ring R constructed from a pair of rings S, T and a bimodule SMT.This gives the relationship of weak Armendarizness between R and S,T,S MT, which plays a very important role in ring theory.In Chapter3, we consider the a-reversibility over which polynomial rings are α-reversible and call them strongly a-reversible rings. A number of properties of this version are established. For an endomorphism α of a ring R, we prove that R is strongly right α-reversible if and only if R[x] is strongly right α-reversible. We next give an example to show that strongly reversible rings need not be strongly a-reversible. For an Armendariz ring R, it is shown that R is right a-reversible if and only if R is strongly right a-reversible if and only if R[x; x-1] is strongly right a-reversible, which is a generalization [25, Proposition2.4]. Let R be a right Ore ring and Q be the classical right quotient ring of R. If R[x] is an a-compatible ring, we prove that R is strongly right α-reversible if and only if Q is strongly right α-reversible. Moreover, we introduce the concept of Nil-a-reversible rings to investigate the nilpotent elements in a-reversible rings.In Chapter4, we introduce the concept of nil-McCoy rings to study the structure of the set of nilpotent elements in McCoy rings. This notion extends the concept of McCoy rings and nil-Armendariz rings. It is shown that every semicommutative ring is nil-McCoy ring, and examples are given to show that nil-McCoy rings need not be semicommutative. More examples of nil-McCoy rings are given by various extensions. For example, it is proved that the direct limit of a direct system of right nil-McCoy rings is also right nil-McCoy. And we show that if I is an nil ideal of a ring R, then R is nil-McCoy if and only if R/I is nil-McCoy. Moreover, we argue about the properties of α-McCoy rings. For a mononiorphism α of a ring R, we prove that if R is weak α-rigid and α-reversible then R is α-McCoy.In Chapter5, the notion of weakly reversible rings is introduced as a non-trivial generalization of reversible rings. Some properties and examples of weakly reversible rings are given. Since every reversible ring is semicommutative [25, Lemma1.4], we may conjecture that weakly reversible rings may be semicomniutative. But we shall give an example to show that there exists a weakly reversible R such that R is not seniicommutative. It is shown that if R is a semicommutative ring, then R[x] is weakly reversible. Furthermore, we investigate the strong reversible properties for an endo- morphism a of a ring R. We do this by considering the strongly reversible property on polynomials in the skew polynomial ring R[x; α] instead of the ring R[x](without skewing the scalar multiplication). This provides us with an opportunity to study strongly reversible rings in a general setting. We prove that R is an a-rigid ring if and only if R is a reduced a-strongly reversible ring. For an endomorphism a of a ring R with δ a α-derivation, we show that if R is an (α,δ)-skew Armendariz ring and satisfies the (α,δ)-compatible condition, then R is reversible if and only if R is (α,δ)-strongly reversible.In Chapter6, we aim to consider reflexive rings and some related rings. We prove that a ring R is a completely reflexive ring if and only if R is a semicommutative reflexive ring. It is shown that:(1) If R is a commutative domain and a is an injective endomorphism of R, then the skewtrivial extension of R by R and a is reflexive.(2) Let R be an algebra over a commutative ring S, and D be the Dorroh extension of R by S. If R is reflexive and S is a domain, then D is reflexive. On the other hand, the reflexivity of some kinds of polynomial rings are investigated. Moreover, the reflexive property of a ring R relative to a given endomorphism a is investigated, which is a generalization of the concept of a reflexive ring and that of an a-rigid ring.