On the Finite Base Property for Semigroup Varieties

No algorithm determines from an arbitrary recursive system of semigroup identities whether the variety defined by this system is finitely based.     

The Structure of Finite c-Local Rings

For a commutative ring R, assume that c is a nonzero element of Z(R) with the property that cZ(R) = {0}. A local ring R is called c-local if Z(R)² = {0, c}, Z(R)³ = {0}, and xZ(R) = {0} implies x ∈ {0, c}. For any finite c-local ring (R, 𝔪), it is proved that the ideal m has a minimal generating set which has a c-partition. The structure and classification up to isomorphism of all finite commutative c-local rings with order greater than 2⁵ are determined.     

On Some Extremal Varieties of Associative Algebras

Suppose that F is a field of prime characteristic p and V _p is the variety of associative algebras over F defined by the identities [[x, y], z] = 0 and x ^p = 0 if p > 2 and by the identities [[x, y], z] = 0 and x ⁴ = 0 if p = 2 (here [x, y] = xy ? yx). As is known, the free algebras of countable rank of the varieties V _p contain non-finitely generated T-spaces. We prove that the varieties V _p are minimal with respect to this property.     

The Structure of Finite Local Principal Ideal Rings

A ring R is called a principal ideal ring (PIR), if each ideal of R is a principal ideal. A local ring (R, 𝔪) is an artinian PIR if and only if its maximal ideal 𝔪 is principal and has finite nilpotency index. In this article, we determine the structure of a finite local PIR.     

On Factorizations of Subformations of One-Generated Saturated Finite Varieties

The following is proved: If 𝔐? is the noncancellable product of the formations 𝔐 and ? and 𝔐? ? 𝔉 for some one-generated saturated finite variety 𝔉, then 𝔐 is soluble.     

On Locally Finite Modular Lattice Varieties of Finite Height

It is shown that the variety generated by a finite modular lattice has only finitely many locally finite covers.      

On Nonfinitely Based Varieties of Groups of Large Prime Exponent

For every sufficiently large odd p, we present a continual set of nonfinitely based varieties of groups of exponent p. The properties of these varieties make it possible to answer some open questions on varieties of groups.     

Infinite Independent Systems of Identities of an Associative Algebra over an Infinite Field of Characteristic Two

Let be the variety of associative (special Jordan, respectively) algebras over an infinite field of characteristic 2 defined by the identity ((((x ₁,x ₂),x ₃), ((x ₄,x ₅),x ₆)), (x ₇,x ₈)) = 0 (((x ₁ x ₂ · x ₃)(x ₄ x ₅ · x ₆))(x ₇ x ₈) = 0, respectively). In this paper, we construct infinite independent systems of identities in the variety ( , respectively). This implies that the set of distinct nonfinitely based subvarieties of the variety has the cardinality of the continuum and that there are algebras in with undecidable word problem.     

The Normalizer Property for Integral Group Rings of Some Finite Nilpotent-by-Nilpotent Groups

In this article, it is shown that the normalizer property holds for the following two kinds of finite nilpotent-by-nilpotent groups: (1) G = NwrH is the standard wreath product of N by H, where N is a finite nilpotent group and H is a finite abelian 2-group; (2) G is a finite group having a normal nilpotent subgroup N such that the integral group ring ?(G/N) has only trivial units. Our results generalize a result of Yuanlin Li and extend some ones obtained by Juriaans, Miranda, and Robério.     

On the Complexity Functions for T-Ideals of Associative Algebras

Let be the sequence of codimension growth for a variety V of associative algebras. We study the complexity function , which is the exponential generating function for the sequence of codimensions. Earlier, the complexity functions were used to study varieties of Lie algebras. The objective of the note is to start the systematic investigation of complexity functions in the associative case. These functions turn out to be a useful tool to study the growth of varieties over a field of arbitrary characteristic. In the present note, the Schreier formula for the complexity functions of one-sided ideals of a free associative algebra is found. This formula is applied to the study of products of T-ideals. An exact formula is obtained for the complexity function of the variety U c of associative algebras generated by the algebra of upper triangular matrices, and it is proved that the function is a quasi-polynomial. The complexity functions for proper identities are investigated. The results for the complexity functions are applied to study the asymptotics of codimension growth. Analogies between the complexity functions of varieties and the Hilbert--Poincaré series of finitely generated algebras are traced.     

有限幂零群通过单群扩张的整群环的正规化子性质

设G是一个有限幂零群通过单群的扩张,即G有一个幂零正规子群N,使得G/N是单群.本文证明了这样的有限群G具有正规化子性质.特别地,内可解群有正规化子性质.     

On Computing Gröbner Bases in Rings of Differential Operators with Coefficients in a Ring

Following the definition of Gr?bner bases in rings of differential operators given by Insa and Pauer (1998), we discuss some computational properties of Gr?bner bases arising when the coefficient set is a ring. First we give examples to show that the generalization of S-polynomials is necessary for computation of Gr?bner bases. Then we prove that under certain conditions the G-S-polynomials can be reduced to be simpler than the original one. Especially for some simple case it is enough to consider S-polynomials in the computation of Gr?bner bases. The algorithm for computation of Gr?bner bases can thus be simplified. Last we discuss the elimination property of Gr?bner bases in rings of differential operators and give some examples of solving PDE by elimination using Gr?bner bases. This work was supported by the NSFC project 60473019.     

Rings of Continuous Functions on Spaces of Finite Rank and the SV Property

Let X be a compact topological space and let C(X) denote the f-ring of all continuous real-valued functions defined on X. A point x in X is said to have rank n if, in C(X), there are n minimal prime ?-ideals contained in the maximal ?-ideal M _x  = {f ? C(X):f(x) = 0}. The space X has finite rank if there is an n ? N such that every point x ? X has rank at most n. We call X an SV space (for survaluation space) if C(X)/P is a valuation domain for each minimal prime ideal P of C(X). Every compact SV space has finite rank. For a bounded continuous function h defined on a cozeroset U of X, we say there is an h-rift at the point z if h cannot be extended continuously to U ∪ {z}. We use sets of points with h-rift to investigate spaces of finite rank and SV spaces. We show that the set of points with h-rift is a subset of the set of points of rank greater than 1 and that whether or not a compact space of finite rank is SV depends on a characteristic of the closure of the set of points with h-rift for each such h. If X has finite rank and the set of points with h-rift is an F-space for each h, then X is an SV space. Moreover, if every x ? X has rank at most 2, then X is an SV space if and only if for each h, the set of points with h-rift is an F-space.     

Varieties of rings,where all subdirectly irreducible finite rings are Armendariz ones

In this paper we study varieties of rings, where all subdirectly irreducible finite rings are Armendariz. We also describe the locally finite varieties of Armendariz rings.     

Varieties and Torsion Classes of m-Groups

We prove that every variety of m-groups is a torsion class; find basis of identities for a product variety of m-groups; and show that the product of every finitely based variety of m-groups and a variety of Abelian m-groups is a finitely based variety.     

The Partition of Unity Method for High-Order Finite Volume Schemes Using Radial Basis Functions Reconstruction

This paper introduces the use of partition of unity method for the develop-ment of a high order finite volume discretization scheme on unstructured grids for solv-ing diffusion models based on partial differential equations. The unknown function and its gradient can be accurately reconstructed using high order optimal recovery based on radial basis functions. The methodology proposed is applied to the noise removal prob-lem in functional surfaces and images. Numerical results demonstrate the effectiveness of the new numerical approach and provide experimental order of convergence.     

Basis Property of Chains of Eigenfunctions for Certain Boundary-Value Problems

Mathematical Notes -     

On free products in varieties of associative algebras

Varieties of associative algebras over a field of characteristic zero are considered. Belov recently proved that, in any variety of this kind, the Hilbert series of a relatively free algebra of finite rank is rational. At the same time, for three important varieties, namely, those of algebras with zero multiplication, of commutative algebras, and of all associative algebras, a stronger assertion holds: for these varieties, formulas that rationally express the Hilbert series of the free product algebra via the Hilbert series of the factors are well known. In the paper, a system of counterexamples is presented which shows that there is no formula of this kind in any other variety, even in the case of two factors one of which is a free algebra. However, if we restrict ourselves to the class of graded PI-algebras generated by their components of degree one, then there exist infinitely many varieties for each of which a similar formula is valid. Translated fromMatematicheskie Zametki, Vol. 65, No. 5, pp. 693–702, May, 1999.