On the relationship between the length of an algebra and the index of nilpotency of its Jacobson radical

A sharp upper bound for the length of an algebra as a function of the index of nilpotency of its Jacobson radical and the length of the quotient algebra by the radical is obtained.     

Bounds for indices of nilpotency and nility

The known bound for the index of nilpotency of a finitely generated nil ring of bounded index is improved and the new bound is applied to obtain other bounds for indices of nilpotency and nility.     

Autour de la condition d'engel dans les algèbres de bernstein

We know that, in general, an algebra satisfying an Engel's condition is a nilalgebra. But an Engel's condition don't implies necessarily the nilpotency of the algebra. In this paper we show that every Bernstein algebra satisfying the second or the third Engel's condition is genetic, that is, the kernel of his weight fonction is nilpotent. This is also the case for a Bernstein algebra satisfying the second weak Engel's condition.

On sait que, en général, toute algèbre vérifiant une condition d'Engel est une nilalgèbre. Cependant, une condition d'Engel n'entraîne pas nécessairement la nilpotence de l'algèbre. Nous montrons, dans ce papier, que taute algèbre de Bernstein vérifiant la deuxième ou la troisième condition d'Engel est génétique, c'est-à-dire, le noyau de sa pondération est nilpotent. C'est aussi le cas pour la deuxième condition faible d'Engel.     

Posets of infinite prinjective type and embeddings of Kronecker modules into the category of prinjective peak I-spaces

For a restricted Lie algebra L over a field of characteristic p > 0 we study the Lie nilpotency index t _L (u(L)) of its restricted universal enveloping algebra u(L). In particular, we determine an upper and a lower bound for t _L (u(L)). Finally, under the assumption that L is p-nilpotent and finite-dimensional, we establish when the Lie nilpotency index of u(L) is maximal.

Communicated by I. Shestakov.     

The length function and matrix algebras

By the length of a finite system of generators for a finite-dimensional associative algebra over an arbitrary field we mean the least positive integer k such that words of length not exceeding k span this algebra (as a vector space). The maximum length for the systems of generators of an algebra is referred to as the length of the algebra. In the present paper, we study the main ring-theoretical properties of the length function: the behavior of the length under unity adjunction, direct sum of algebras, passage to subalgebras and homomorphic images. We give an upper bound for the length of the algebra as a function of the nilpotency index of its Jacobson radical and the length of the quotient algebra. We also provide examples of length computation for certain algebras, in particular, for the following classical matrix subalgebras: the algebra of upper triangular matrices, the algebra of diagonal matrices, the Schur algebra, Courter’s algebra, and for the classes of local and commutative algebras.     

Finiteness of the standard basis of a T-ideal containing lie nilpotency of index 4

Recently the author presented a notion of standard basis for a T-ideal of the free associative algebra over a field of zero characteristic. It was proved that it is finite if the T-ideal contains either Lie nilpotency of index 3 or a multilinear product of commutators of degree 2. Here we prove the finiteness of the reduced standard basis of any T-ideal containing Lie nilpotency of index 4.     

<Emphasis Type="Italic">n</Emphasis>-Tuple algebras of associative type

We study properties of n-tuple algebras of associative type. We show that the nilpotency of an n-tuple algebra of associative type is determined by the nilpotency of each element. In addition, we characterize the nilpotency of an n-tuple algebra of associative type in terms of the trace function. In the final part of the paper, we show that a homogeneously semisimple n-tuple algebra of associative type is the direct sum of two-sided ideals each of which is a homogeneously simple n-tuple algebra of associative type.     

The p-Modular Descent Algebra of the Symmetric Group

The descent algebra of the symmetric group, over a field ofnon-zero characteristic p, is studied. A homomorphism into thealgebra of generalised p-modular characters of the symmetricgroup is defined. This is then used to determine the radical,and its nilpotency index. It also allows the irreducible representationsof the descent algebra to be described. 1991 Mathematics SubjectClassification 20F32.     

广义模糊幂零矩阵

路代数是加法幂等的半环,它包括了布尔代数,模糊代数,分配格及斜坡.因此布尔矩阵,模糊矩阵,格矩阵及斜矩阵都是路代数上的典型矩阵.广义模糊幂零矩阵指的就是路代数上的幂零矩阵.在2010年,Tan研究了路代数上矩阵的幂零性.在Tan的基础上继续讨论了路代数上幂零矩阵的幂零指数.     

BOUNDS FOR INDICES OF NILPOTENCY OF FINITELY GENERATED NIL RINGS OF FINITE INDEX

Abstract

An improved bound is given for the index of nil-potency of a finitely generated nil ring of index n in terms of the index of nilpotency of the ideal generated by T^m where m = [n/2] and T is a m-subset of the set of generators. If m = 3 it is proved that T¹⁰ is contained in an ideal generated by twenty-seven cubes and this is applied to get bounds for the index of nilpotency of a finitely generated nil ring of index 6 or 7, bounds which are less than one hundredth of the bounds we obtained in a previous paper.     

Nil and strongly nil derivations

In a semiprime algebra R over a field of characteristic p ≥ 0, a derivation d is called nil if for each x ? R there is n such that dⁿx = 0. It is called strongly nil if it is induced by a nil element in the symmetric ring of quotients. The first result of this paper is an intrinsic criterion on a nil derivation being strongly nil. We then use this criterion to establish some relation between a nilpotent derivation and a strongly nilpotent derivation after a preliminary discussion of the nilpotency, or index of nilpotency, of a nilpotent derivation. Examples are given later in the paper to show that in some sense these results are best possible. This paper is self contained.     

Solvability of the ideal of all weight zero elements in bernstein algebras

We use a computer to verify that the ideal N of all weight zero elements of any (not necessarily finite dimensional) Bernstein algebra is solvable of index ≤4. We also use a computer to verify that N ² is nilpotent of index ≤9. We give three examples of Bernstein algebras which show that various hypotheses like finite dimensionality, finitely generatedA ² = A, are separately not enough to force N to be nilpotent.     

Invariance of dimension of p-subspaces in bernstein algebras *

The purpose of this paper is to prove that some vector subspaces, called p-subspaces, obtained from the Peirce decomposition of a Bernstein algebra A relative to an idempotent have dimensions which are independent of the idempotent used to decompose A. In particular, for Bernstein-Jordan algebras, this fact is true for every such subspace and this implies that all p-subspaces of a Bernstein algebra, contained in V, for A = Ke + U + V, have invariant dimension. Finally we classify all p-subspaces of degree ≥ 3, contained in U, in a Bernstein algebra A, relative to the invariance (or not) of dimension.     

Nilpotency of the Alternator Ideal of a Finitely Generated Binary (-1,1)-Algebra

We prove nilpotency of the alternator ideal of a finitely generated binary (-1,1)-algebra. An algebra is a binary (-1,1)-algebra if its every 2-generated subalgebra is an algebra of type (-1,1). While proving the main theorem we obtain various consequences: a prime finitely generated binary (-1,1)-algebra is alternative; the Mikheev radical of an arbitrary binary (-1,1)-algebra coincides with the locally nilpotent radical; a simple binary (-1,1)-algebra is alternative; the radical of a free finitely generated binary (-1,1)-algebra is solvable. Moreover, from the main result we derive nilpotency of the radical of a finitely generated binary (-1,1)-algebra with an essential identity.     

The rational homotopy Lie algebra of classifying spaces for formal two-stage spaces

We compute the center and nilpotency of the graded Lie algebra for a large class of formal spaces X. The latter calculation determines the rational homotopical nilpotency of the space of self-equivalences aut₁(X) for these X. Our results apply, in particular, when X is a complex or symplectic flag manifold.     

The modular group algebras of p-groups of maximal class II

The modular isomorphism problem is settled for 3-groups of maximal class but two families of groups. Moreover, the conjecture that the ideals belonging to the lower central series of a group base are determined by the structure of the group algebra is refuted in greatest generality by virtue of a single group of order 81 of maximal class, and it is proved that the nilpotency class is determined by the structure of the group algebra for p-groups of maximal class.     

李color代数的T~*-扩张

介绍了李color代数的T*-扩张的定义,并证明李color代数的很多性质,如幂零性、可解性和可分解性,都可以提升到它的T*-扩张上.还证明在特征不等于2的代数闭域上,有限维幂零二次李color代数A等距同构于一个幂零李color代数B的T*-扩张,并且B的幂零长度不超过A的一半.此外,用上同调的方法研究了李color代数的T*-扩张的等价类.     

The Coexponent of a Finite p-Group

Abstract

In this paper,we present a sharp bound for the nilpotency class of a finite p-group (where p is an odd prime) in terms of its coexponent. As to a powerful p-group,we give the sharp bound for the nilpotency class in terms of its coexponent for arbitrary prime p.     

The Nilpotency Properties of the Leibniz Algebra Mn(C)D

We study the nilpotency properties of the Leibniz algebras constructed by means of D-mappings on the algebra of complex square matrices M _n (C). In particular, we obtain a criterion for nilpotency of these algebras in terms of the properties of a D-mapping. We prove also that the Leibniz algebras under consideration cannot be simple.