Ore Extensions Research Articles

Ore Extensions over Total Valuation Rings

It is shown that any Ore extension R = V[x;σ,δ] over a total valuation ring V is always v-Bezout which is a generalization of commutative GCD domains. By using this result, a necessary and sufficient condition are given for R to be fully left bounded.

Normal Quadratics in Ore Extensions, Quantum Planes, and Quantized Weyl Algebras

Let R be a Noetherian domain and let (σ,δ) be a quasi-derivation of R such that σ is an automorphism. There is an induced quasi-derivation on the classical quotient ring Q of R. Suppose F=t 2−v is normal in the Ore extension R[t;σ,δ] where v∈R. We show F is prime in R[t;σ,δ] if and only if F is irreducible in Q[t;σ,δ] if and only if there does not exist w∈Q such that v=σ(w)w−δ(w). We apply this result to classify prime quadratic forms in quantum planes and quantized Weyl algebras.

Quotient Rings of Ore Extensions with More than One Indeterminate

Let R be a domain and R[X; D] the Ore extension of R by a sequence D of derivations of R. If D has length ≥ 2, we show that the symmetric Utumi quotient ring of R[X; D] is U s (R)[X; D], where U s (R) is the symmetric Utumi quotient ring of R. Consequently, X-inner automorphisms of R[X; D] are induced by units of U s (R) and the extended centroid of R[X; D] consists of those elements α in the center of U s (R) such that δ(α) = 0 for all δ ∈ D. These extend the known results for free algebras.

Double ore extensions

A double Ore extension is a natural generalization of the Ore extension. We prove that a connected graded double Ore extension of an Artin–Schelter regular algebra is Artin–Schelter regular. Some other basic properties such as the determinant of the DE-data are studied. Using the double Ore extension, we construct 26 families of Artin–Schelter regular algebras of global dimension four in a sequel paper.

Annihilator Conditions on Ore Extensions

In this paper, we extend the study of various annihilator conditions on the nearring of polynomials and the nearring of formal power series to skew polynomials and skew formal power series in which addition and substitution are used as operations. A result of Hong et al. on the skew polynomial rings and skew power series of an α-rigid Baer ring is extended to Baer conditions in the nearrings of skew polynomials and skew formal power series. Also, for an injective endomorphism α of a ring R, it is shown that R is α-rigid if and only if the nearring (R[x; α], +, ◦) is reduced, if and only if the nearring (R0[[x; α]], +, ◦) is reduced, if and only if R is a reduced and near Armendariz ring.

ON ORE EXTENSIONS OF QUASI-BAER RINGS

A ring R is called (right principally) quasi-Baer if the right annihilator of every (principal right) ideal of R is generated by an idempotent. We study on the relationship between the quasi-Baer and p.q.-Baer property of a ring R and these of the Ore extension R[x; α, δ] for any automorphism α and α-derivation δ of R.

Ore Extensions of Skew Armendariz Rings

Let α be an endomorphism and δ an α-derivation of a ring R. We introduce the notion of skew-Armendariz rings which are a generalization of α-skew Armendariz rings and α-rigid rings and extend the classes of non reduced skew-Armendariz rings. Some properties of this generalization are established, and connections of properties of a skew-Armendariz ring R with those of the Ore extension R[x; α, δ] are investigated. As a consequence we extend and unify several known results related to Armendariz rings.

Generalizing the notion of Koszul algebra

We introduce a generalization of the notion of a Koszul algebra, which includes graded algebras with relations in different degrees, and we establish some of the basic properties of these algebras. This class is closed under twists, twisted tensor products, regular central extensions and Ore extensions. We explore the monomial algebras in this class and we include some well-known examples of algebras that fall into this class.

Ore extensions which are GPI-rings

Let R be a prime ring and δ a σ-derivation of R, where σ is an automorphism of R. It is proved that the skew polynomial ring $$R[t; \sigma, \delta]$$ is a GPI-ring (PI-ring resp.) if and only if R is a GPI-ring (PI-ring resp.), δ is quasi-algebraic, and σ is quasi-inner. If $$R[t; \sigma, \delta]$$ is a GPI-ring then soc $$(Q[t;\sigma,\delta]\tilde C) = \Big({\rm soc}(Q)[t;\sigma,\delta]\Big)\tilde C$$ , where Q is the symmetric Martindale quotient ring of R and where $$\tilde C$$ denotes the extended centroid of $$Q[t; \sigma, \delta]$$ . If $$R[t; \sigma, \delta]$$ is a PI-ring, its PI-degree is determined as follows: $$(1) {\rm PI-deg}(R[t; \sigma, \delta]) = {\rm PI-deg}(R) \times {\rm out-deg}(\delta)$$ if δ is X-outer, and $$(2) {\rm PI-deg}(R[t; \sigma, \delta]) = {\rm PI-deg}(R) \times {\rm out-deg}(\sigma)$$ if δ is X-inner.

Constructing Clifford quantum [formula omitted]s with finitely many points

We present an algebro-geometric technique for constructing regular Clifford algebras A of global dimension four with associated point scheme consisting of a prespecified finite number of points. In particular, if A has more than one point in its point scheme, then the number of points in the point scheme can be obtained from the number of intersection points of two planar cubic divisors; these cubic divisors correspond to regular Clifford subalgebras of A of global dimension three. If A has exactly a finite number, n, of distinct points in its point scheme, then n ∈ { 1 , 3 , 4 , 5 , … , 13 , 14 , 16 , 18 , 20 } and all these possibilities occur. We also prove that if a regular Clifford algebra R of global dimension d ⩾ 2 has exactly a finite number, n, of distinct isomorphism classes of point modules, then n is odd if and only if R is an Ore extension of a regular Clifford subalgebra of R of global dimension d − 1 .

Jacobson Radicals of Ore Extensions of Derivation Type

Let D be a sequence of derivations of a ring R. We form the Ore extension R[X; D]. Let 𝒩(R) denote the nil radical of R and 𝒥(R[X; D]), the Jacobson radical of R[X; D]. We show that 𝒩(R) = 0 implies 𝒥(R[X; D]) = 0 under one of the following conditions: (1) R is a PI-ring. (2) One of derivations in D is quasi-algebraic. (3) R satisfies the ascending chain condition on right annihilators.

Ore extensions over pseudo-valuation rings

Let R be a commutative Noetherian Q-algebra (Q is the field of rational numbers). Let δ be a derivation of R and σ be an automorphism of R. Then we prove the following: 1. If R is a Pseudo-valuation ring, then R[x, δ] is also a Pseudo-valuation ring. 2. If R is a divided ring, then R[x, δ] is also a divided ring. 3. If R is a Pseudo-valuation ring, thenR[x, x−1, σ] is also a Pseudo-valuation ring. 4. If R is a divided ring, then R[x, x−1, σ] is also a divided ring. Mathemmatics Subject Classification: Primary 16-XX; Secondary 16P40, 16P50, 16U20

ORE EXTENSIONS SATISFYING A POLYNOMIAL IDENTITY

Necessary and sufficient conditions for an Ore extension S = R[x;σ,δ] to be a PI ring are given in the case σ is an injective endomorphism of a semiprime ring R satisfying the ACC on annihilators. Also, for an arbitrary endomorphism τ of R, a characterization of Ore extensions R[x;τ] which are PI rings is given, provided the coefficient ring R is noetherian.

NON-COMMUTATIVE DUPLICATES OF FINITE SETS

We classify the twisted tensor products of a finite set algebra with a two elements set algebra using colored quivers obtained through considerations analogous to Ore extensions. This provides also a classification of entwining structures between a finite set algebra and the grouplike coalgebra on two elements. The resulting 2-nilpotent algebras have particular features with respect to Hochschild (co)homology and cyclic homology.

Generalized orderings and rings of fractions

Every total ordering of a commutative domain can be extended uniquely to its field of fractions. This result is extended in two directions. Firstly, the notion of a total ordering is generalized so that a nonzero element can have more than two signs (in fact, these signs form a group). Secondly, commutative domains are replaced by noncommutative ones and we consider the following types of rings of fractions: Ore extensions, maximal (right or two-sided) rings of fractions, division hulls of free algebras and epic fields. Throughout the paper several examples are given to illustrate the theory.

Ore extensions of zip and reversible rings

Ore extensions of zip and reversible rings

Goldie Conditions for Ore Extensions over Semiprime Rings

Let R be a ring, σ an injective endomorphism of R and δ a σ-derivation of R. We prove that if R is semiprime left Goldie then the same holds for the Ore extension R[x;σ,δ] and both rings have the same left uniform dimension.

Rigid Ideals and Radicals of Ore Extensions

For an endomorphism σ of a ring R, Krempa called σ a rigid endomorphism if aσ(a)=0 implies a=0 for a∈R. A ring R is called rigid if there exists a rigid endomorphism of R. In this paper, we extend the σ-rigid property of a ring R to an ideal of R. For a σ-ideal I of a ring R, we call I a σ-rigid ideal if aσ(a)∈I implies a∈I for a∈R. We characterize σ-rigid ideals and study related properties. The connections of the prime radical and the upper nil radical of R with the prime radical and the upper nil radical of the Ore extension R[x;σ,δ], respectively, are also investigated.

Ore extensions over duo rings

We show that there exist noncommutative Ore extensions in which every right ideal is two-sided. This answers a problem posed by Marks in [G. Marks, Duo rings and Ore extensions, J. Algebra 280 (2) (2004) 463–471]. We also provide an easy construction of one-sided duo rings.

Generic Lie colour algebras

We describe a type of Lie colour algebra, which we call generic, whose universal enveloping algebra is a domain with finite global dimension. Moreover, it is an iterated Ore extension. We provide an application and show Gröbner basis methods can be used to study universal enveloping algebras of factors of generic Lie colour algebras.