A Ring Variety without an Independent Basis

An example of a ring variety without an independent basis is constructed. It is proved that this variety is the intersection of two independently based varieties.     

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.     

Identities of Semigroups of Triangular Matrices over Finite Fields

It is proved that the semigroup of all triangular n × n matrices over a finite field K is inherently nonfinitely based if and only if n > 3 and |K|> 2.     

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.     

中心代数上的矩阵方程组

设Ω是一个特征非2的具有对合反自同构的有限维中心代数.本文研究Ω上的两个矩阵方程组,分别给出了其有一般解和次(斜)自共轭解的充要条件.     

Infinite independent systems of the identities of the associative algebra over an infinite field of characteristic <Emphasis Type="Italic">p</Emphasis> > 0

In this paper some infinitely based varieties of groups are constructed and these results are transferred to the associative algebras (or Lie algebras) over an infinite field of an arbitrary positive characteristic.