On the semigroup nilpotency and the Lie nilpotency of associative algebras

To each associative ringR we can assign the adjoint Lie ringR ⁽⁻⁾ (with the operation(a,b)=ab−ba) and two semigroups, the multiplicative semigroupM(R) and the associated semigroupA(R) (with the operationaob=ab+a+b). It is clear that a Lie ringR ⁽⁻⁾ is commutative if and only if the semigroupM(R) (orA(R)) is commutative. In the present paper we try to generalize this observation to the case in whichR ⁽⁻⁾ is a nilpotent Lie ring. It is proved that ifR is an associative algebra with identity element over an infinite fieldF, then the algebraR ⁽⁻⁾ is nilpotent of lengthc if and only if the semigroupM(R) (orA(R)) is nilpotent of lengthc (in the sense of A. I. Mal'tsev or B. Neumann and T. Taylor). For the case in whichR is an algebra without identity element overF, this assertion remains valid forA(R), but fails forM(R). Another similar results are obtained. Translated fromMatematicheskie Zametki, Vol. 62, No. 4, pp. 510–519, October, 1997. Translated by A. I. Shtern     

The residual nilpotency of the augmentation ideal and the residual nilpotency of some classes of groups

The following theorem is proved: LetG be any group. Then the augmentation ideal ofZG is residually nilpotent if and only ifG is approximated by nilpotent groups without torsion or discriminated by nilpotent p_i,-groups,i ∈I, of finite exponents. This theorem is applied to obtain conditions under which the groupsF/N′ are residually nilpotent whereF is a free non-cyclic group and N?F.     

Third-order nilpotency, nice reachability and asymptotic stability

We consider an affine control system whose vector fields span a third-order nilpotent Lie algebra. We show that the reachable set at time T using measurable controls is equivalent to the reachable set at time T using piecewise-constant controls with no more than four switches. The bound on the number of switches is uniform over any final time T. As a corollary, we derive a new sufficient condition for stability of nonlinear switched systems under arbitrary switching. This provides a partial solution to an open problem posed in [D. Liberzon, Lie algebras and stability of switched nonlinear systems, in: V. Blondel, A. Megretski (Eds.), Unsolved Problems in Mathematical Systems and Control Theory, Princeton Univ. Press, 2004, pp. 203-207].     

Bounds on the indices of the spectral-circle eigenvalues of a nonnegative matrix

Let P be a square, nonnegative matrix. A set of k classes of P is called a chain of length k if every class in the set has access to or from any other class in the set. A chain is called a λ-chain if λ is in the spectrum of each submatrix of P corresponding to a class in the chain. Our main result states that the index of a complex number λ lying on the spectral circle of P is bounded above by the length of the longest λ-chain. We illustrate, by examples, that this bound can be attained but could sometimes be arbitrarily poor.     

A sufficient condition for nilpotency of the commutator subgroup

Let G be a finite group with the property that if a and b are commutators of coprime orders, then |ab| = |a||b|. We show that G′ is nilpotent.     

伪Smarandache函数的上下界

对于正整数n,设Z(n)=min{m|m∈N,1/2m(m+1)≡0(modn)},称为n的伪Smarandache函数.设r是正整数.根据广义Ramanujan-Nagell方程的结果,运用初等数论方法证明了下列结果:i)1/2(-1+(8n+1)≤Z(n)≤2n-1.ii)当r≠1,2,3或5时,Z(2~r+1)≥1/2(-1+(2~(r+3)·5+41)).iii)当r≠1,2,3,4或12时,Z(2~r-1)≥1/2(-1+(2~(r+3)·3-23).     

On the n—nilpotency

In this paper we prove that, for every integer n ≥ 1, d ≥ m ≥ 2, the sum of m n- nilpotent ideals of an algebra may not be d- nilpotent. This leads to a similar result in group theory; the product of m n- nilpotent normal subgroups may not be d- nilpotent. We apply these results to solve Rozhkov’s question by constructing some n- finite, infinite p- groups generated by n + 1 conjugates.     

On the nilpotency and decomposition of Lie-type algebras

In the paper, an analog of the Engel theorem for graded algebras admitting a Lie-type module is proved. Moreover, it is shown that every semisimple algebra of associative type with ordered grading and one-dimensional grading subspaces is the direct sum of two-sided ideals that are simple algebras.     

Cardinality and nilpotency of localizations of groups andG-modules

We consider the effect of a coagumented idempotent functorJ in the the category of groups orG-modules whereG is a fixed group. We are interested in the ‘extent’ to which such functors change the structure of the objects to which they are applied. Some positive results are obtained and examples are given concerning the cardinality and structure ofJ(A) in terms of the cardinality and structure ofA, where the latter is a torsion abelian group. For non-abelian groups some partial results and examples are given connecting the nilpotency classes and the varieties of a groupG andJ(G). Similar but stronger results are obtained in the category ofG-modules.