One-Sided Ideal Growth of Free Associative Algebras

Let R be a finitely generated associative algebra with unity over a finite field . Denote by a _n (R) the number of left ideals J ⊂ R such that dim R/J = n for all n ≥ 1. We explicitly compute and find asymptotics of the left ideal growth for the free associative algebra A _d of rank d with unity over , where d ≥ 1. This function yields a bound a _n (R) ≤ a _n (A _d ), , where R is an arbitrary algebra generated by d elements. Denote by m _n (R) the number of maximal left ideals J ⊂ R such that dim R/J = n, for n ≥ 1. Let d ≥ 2, we prove that m _n (A _d ) ≈ a _n (A _d ) as n → ∞.     

Free Associative Algebras

On page 37, lines 7–8, for "principal ideal domain" read"Euclidean domain".     

The Inverse Function Theorem for Free Associative Algebras

We obtain a necessary and sufficient condition for a given collection of elements to freely generate a free associative algebra. We present some necessary conditions for primitivity of an element of a free associative algebra of rank 2.     

Codimension Growth of Strong Lie Nilpotent Associative Algebras

We study Lie nilpotent varieties of associative algebras. We explicitly compute the codimension growth for the variety of strong Lie nilpotent associative algebras. The codimension growth is polynomial and found in terms of Stirling numbers of the first kind. To achieve the result we take the free Lie algebra of countable rank L(X), consider its filtration by the lower central series and shift it. Next we apply generating functions of special type to the induced filtration of the universal enveloping algebra U(L(X)) = A(X).     

On Codimension Growth of Finitely Generated Associative Algebras

LetAbe a PI-algebra over a fieldF. We study the asymptotic behavior of the sequence of codimensionsc_n(A) ofA. We show that ifAis finitely generated overFthenInv(A)=lim_n→∞ always exists and is an integer. We also obtain the following characterization of simple algebras:Ais finite dimensional central simple overFif and only ifInv(A)=dim=A.     

Jordan Gradings on Associative Algebras

In this paper we apply the method of functional identities to the study of group gradings by an abelian group G on simple Jordan algebras, under very mild restrictions on the grading group or the base field of coefficients.     

On Strongly Associative Group Algebras

Given a partial action α of a group G on the group algebra FH, where H is a finite group and F is a field whose characteristic p divides the order of H, we investigate the associativity question of the partial crossed product FH*_α G. If FH*_α G is associative for any G and any α, then FH is called “strongly associative.” Using a result of Dokuchaev and Exel (2005 Dokuchaev , M. , Exel , R. ( 2005 ). Associativity of crossed products by partial actions, enveloping actions and partial representations . Trans. Amer. Math. Soc. 357 : 1931 – 1952 .[Crossref], [Web of Science ®] , [Google Scholar]) we characterize the strongly associative modular group algebras FH for several classes of groups H.     

Existence of Gradings on Associative Algebras

In this article, we study the existence of gradings on finite dimensional associative algebras. We prove that a connected algebra A does not have a nontrivial grading if and only if A is basic, its quiver has one vertex, and its group of outer automorphisms is unipotent. We apply this result to prove that up to graded Morita equivalence there do not exist nontrivial gradings on the blocks of group algebras with quaternion defect groups and one isomorphism class of simple modules.     

Degenerations of Representations of Associative Algebras

The purpose of this note is to give a fast introduction to some problems of homological and geometrical nature related to finite-dimensional representations of finitely generated, and especially, finite-dimensional algebras over a field. Some of these results can also be extended to the situation where the field is not algebraically closed, and some of the results can even be extended to the situation where one is considering algebras over a commutative artin ring. For the results which hold true in the most general situation the proofs become most elegant since they depend on using length arguments only and thereby forgetting about the nature of a field altogether. Received: July 2007     

Some Classes of Nilpotent Associative Algebras

We further the study of rings with no middle class by focusing on an interpretation of that property in terms of the lattice of hereditary pretorsion classes over a given ring. For non-semisimple rings, the absence of a middle class is equivalent to the requirement that the class of all semisimple right modules be a coatom in that lattice. Taking advantage of this perspective, we discover new facts and shed light on others already known with a possibly more direct interpretation without having to refer to an exhaustive analysis of the structure theorems available in the literature. Our approach also allows us to characterize rings with no middle class in terms of hereditary pretorsion classes containing the class of all singular right modules. We discuss the open problem of whether there is a ring with no right middle class which is not right Noetherian and see, in particular, that an indecomposable ring satisfying that property would have to be Morita equivalent to a certain type of subring of a full linear ring.     

Four-Dimensional Real Third-Power Associative Division Algebras

We study third-power associative division algebras A over a field 𝕂 of characteristic different from 2. Those algebras having dimension ≤2 are commutative. When 𝕂 is the field ? of real numbers, those algebras having dimension 4 are power-commutative in each of the following two cases:

1. A contains a central element;

2. A satisfies the additional identity (x, x³, x) = 0.

     

Ideal Structure of Clifford Algebras

The structures of the ideals of Clifford algebras which can be both infinite dimensional and degenerate over the real numbers are investigated.      

Ideal Spaces of Banach Algebras

The ideal space Id(A) of a Banach algebra A is studied as abitopological space Id(A), _u, _n, where _u is the weakest topologyfor which all the norm functions I || a + I|| (with a A andI Id(A)) are upper semi-continuous, and _n is the de Groot dualof _u. When A is separable, _nu is either a compact, metrizabletopology, or it is neither Hausdorff nor first countable. TAF-algebrasare shown to exhibit the first type of behaviour. Applicationsto Banach bundles (which motivate the study), and to PI-Banachalgebras, are given. 1991 Mathematics Subject Classification:46H10, 46J20.     

Graded Associative Conformal Algebras of Finite Type

In this paper, we consider graded associative conformal algebras. The class of these objects includes pseudo-algebras over non-cocommutative Hopf algebras of regular functions on some linear algebraic groups. In particular, an associative conformal algebra which is graded by a finite group Γ is a pseudo-algebra over the coordinate Hopf algebra of a linear algebraic group G such that the identity component G ⁰ is the affine line and G/G ⁰???Γ. A classification of simple and semisimple graded associative conformal algebras of finite type is obtained.