Exceptional prime alternative algebras

We prove that every exceptional prime alternative algebra satisfies the Engel identity [[x, y], y] = 0 of index 2.     

Algebraic prime subalgebras in simple Artinian algebras

We prove a Wedderburn-Artin type theorem for algebraic prime subalgebras in simple Artinian algebras, giving a generalized version of Yahaghi’s theorem [B.R. Yahaghi, On F-algebras of algebraic matrices over a subfield F of the center of a division ring, Linear Algebra Appl. 418 (2006) 599-613]. We also show that every semiprime left algebraic subring in a semiprime right Goldie ring must be a semiprime Artinian ring.     

Fuzzy prime filters of lattice implication algebras

In this paper, some properties of fuzzy filters are given. Besides, the structure of fuzzy filters are further studied. And finally, the concept of fuzzy prime filter is proposed with some equivalent conditions of fuzzy prime filters obtained.     

The graded Gelfand--Kirillov dimension of verbally prime algebras

Let E be the infinite-dimensional Grassmann algebra over a field F of characteristic 0. In this article, we consider the verbally prime algebras M _n (F), M _n (E) and M _a,b (E) endowed with their gradings induced by that of Vasilovsky, and we compute their graded Gelfand--Kirillov dimensions.     

Growth of central polynomials of verbally prime algebras

We compute the exact asymptotics of the codimension sequence for the central polynomials of k × k matrices and show that it is asymptotic to \(\frac{1}{k^2}\) times the ordinary cocharacter. For the other verbally prime algebras we show that these sequences are bounded above and below by constants times the ordinary codimensions.     

Cyclicity conditions for division algebras of prime degree

Let be a division algebra of prime degree . A set of criteria is given for cyclicity of in terms of subgroups of the multiplicative group of . It is essentially shown that is cyclic if and only if contains a nonabelian metabelian subgroup.

     

Division algebras of prime degree with infinite genus

The genus gen(D) of a finite-dimensional central division algebra D over a field F is defined as the collection of classes [D′] ∈ Br(F), where D′ is a central division F-algebra having the same maximal subfields as D. For any prime p, we construct a division algebra of degree p with infinite genus. Moreover, we show that there exists a field K such that there are infinitely many nonisomorphic central division K-algebras of degree p and any two such algebras have the same genus.     

Bicyclic algebras of prime exponent over function fields

We examine some properties of bicyclic algebras, i.e. the tensor product of two cyclic algebras, defined over a purely transcendental function field in one variable. We focus on the following problem: When does the set of local invariants of such an algebra coincide with the set of local invariants of some cyclic algebra? Although we show this is not always the case, we determine when it happens for the case where all degeneration points are defined over the ground field. Our main tool is Faddeev's theory. We also study a geometric counterpart of this problem (pencils of Severi-Brauer varieties with prescribed degeneration data).

     

Maximal and minimal prime ideals of incidence algebras

Abstract

In this paper we study the relationship between some homological properties such as the weak and the global dimension, the strong n-coherence, and the (n, d)-property, where n and d are two integers, of a commutative ring and its subrings retract. A special application is the transfer of these properties from a commutative ring to its fixed subring with respect to a subgroup of its group of automorphisms. It concludes with a discussion of the scopes and limits of our results.     

Distributive and Semihereditary Rings

Let A be a right and left distributive ring. For a positive integer n, we obtain a criterion of projectivity of all n-generated right ideals of the ring A and a criterion of the right semi-heredity of the ring A.__________Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 1, pp. 253–258, 2003.     

On Semihereditary Maximal Orders

Let A be an order integral over a valuation ring V in a centralsimple F-algebra, where F is the fraction field of V. We showthat (a) if (V^h, F^h) is the Henselization of (V, F), then Ais a semihereditary maximal order if and only if A_VV^h is a semihereditarymaximal order, generalizing the result by Haile, Morandi andWadsworth, and (b) if J(V) is a principal ideal of V, then asemihereditary V-order is an intersection of finitely many conjugatesemihereditary maximal orders; if not, then there is only onemaximal order containing the V-order. 1991 Mathematics SubjectClassification 16H05.     

Links between cofinite prime ideals in quantum function algebras

We describe the skew primitive elements in a multiparameter enveloping algebraU=U _q,p ⁻¹ (g) and the links between cofinite maximal ideals in the corresponding quantum function algebra ℂ _q [G]. These results are applied to determine the coradical filtration forU, and to obtain a moduli space for multiparameter Drinfeld doubles. Research partially supported by NSA grant MDA 904-93-H3016.     

Chains of prime ideals in tensor products of algebras

This paper investigates the length of particular chains of prime ideals in tensor products of algebras over a field k. As an application, we compute dim(A⊗_kA) for a new family of domains A that are k-algebras.