Rings whose nilpotents form a multiplicative set

Nilpotents in Armendariz and abelian π-regular rings are multiplicatively closed. However, this fact need not hold in many kinds of rings. This article concerns a class of rings whose nilpotents are closed under multiplication. Rings contained in this class are said to be nilpotent-closed, and the structure of nilpotent-closed rings is investigated in relation with various situations which happen ordinarily in the study of noncommutative ring theory. In the procedure, various sorts of rings are investigated, so that they are nilpotent-closed. We also study familiar conditions under which nilpotents in nilpotent-closed rings form a subring.     

Fano contact manifolds and nilpotent orbits

A contact structure on a complex manifold M is a corank 1 subbundle F of T_M such that the bilinear form on F with values in the quotient line bundle L = T_M/F deduced from the Lie bracket of vector fields is everywhere non-degenerate. In this paper we consider the case where M is a Fano manifold; this implies that L is ample.?If is a simple Lie algebra, the unique closed orbit in (for the adjoint action) is a Fano contact manifold; it is conjectured that every Fano contact manifold is obtained in this way. A positive answer would imply an analogous result for compact quaternion-Kahler manifolds with positive scalar curvature, a longstanding question in Riemannian geometry.?In this paper we solve the conjecture under the additional assumptions that the group of contact automorphisms of M is reductive, and that the image of the rational map M P(H ₀(M, L)*) sociated to L has maximum dimension. The proof relies on the properties of the nilpotent orbits in a semi-simple Lie algebra, in particular on the work of R. Brylinski and B. Kostant. Received: July 28, 1997     

Semigroups generated by nilpotent transformations

In a seminal paper published in 1966, John Howie characterised the elements of _x, the semigroup (under composition) of all total transformations of a set X into itself, which can be written as a product of idempotents in _x. We now initiate the study of the subsemigroup of _x, the semigroup of all partial transformations of X, which is generated by the nilpotents of _x     

Concentrated matrix Langevin distributions

This paper concerns the matrix Langevin distributions, exponential-type distributions defined on the two manifolds of our interest, the Stiefel manifold V_k,m and the manifold P_k,m−k of m×m orthogonal projection matrices idempotent of rank k which is equivalent to the Grassmann manifold G_k,m−k. Asymptotic theorems are derived when the concentration parameters of the distributions are large. We investigate the asymptotic behavior of distributions of some (matrix) statistics constructed based on the sample mean matrices in connection with testing hypotheses of the orientation parameters, and obtain asymptotic results in the estimation of large concentration parameters and in the classification of the matrix Langevin distributions.     

On the idempotents,nilpotents, units and zero-divisors of finite rings

In this article, first we find the number of idempotents and the zero-divisors of a matrix ring over a finite field F. Next, given the order of the Jacobson radical of the finite unital ring R, we find the number of units, nilpotents and zero-divisors of R. Moreover, we find an upper bound for the number of idempotents of a finite ring which is in general smaller than the upper bound found by MacHale [Proc. R. Ir. 1982;82A(1):9–12]. Finally, we find the above-mentioned numbers in some matrix rings and quaternion rings.     

Sur les sous-groupes nilpotents du groupe de Cremona

Résumé. Nous décrivons les sous-groupes nilpotents non virtuellement abéliens du groupe des transformations birationnelles du plan projectif complexe: un tel groupe est d'ordre fini ou virtuellement métabélien.      

On the structure of a Morse form foliation

The foliation of a Morse form ω on a closed manifold M is considered. Its maximal components (cylinders formed by compact leaves) form the foliation graph; the cycle rank of this graph is calculated. The number of minimal and maximal components is estimated in terms of characteristics of M and ω. Conditions for the presence of minimal components and homologically non-trivial compact leaves are given in terms of rk ω and Sing ω. The set of the ranks of all forms defining a given foliation without minimal components is described. It is shown that if ω has more centers than conic singularities then b ₁(M) = 0 and thus the foliation has no minimal components and homologically non-trivial compact leaves, its folitation graph being a tree.      

Boundedness and Complete Distributivity

We extend the concept of constructive complete distributivity so as to make it applicable to ordered sets admitting merely bounded suprema. The KZ-doctrine for bounded suprema is of some independent interest and a few results about it are given. The 2-category of ordered sets admitting bounded suprema over which inhabited (classically non-empty) infima distribute is shown to be bi-equivalent to a 2-category defined in terms of idempotent relations. As a corollary we obtain a simple construction of the non-negative reals.     

Homology and cohomology in hypo-analytic structures of the hypersurface type

The work is concerned with local exactness in the cohomology of the differential complex associated with a hypo-analytic structure on a smooth manifold. Only structures of the hypersurface type are considered, i.e., structures in which the rank of the characteristic set does not exceed one. Among them are the CR structures of real hypersurfaces in a complex manifold. The main theorem states anecessary condition for local exactness in dimensionq to hold. The condition is stated in terms of the natural homology associated with the differential complex, as inherited by the level sets of the imaginary part of an arbitrary solutionw whose differential spans the characteristic set at the central point. An intersection number, which generalizes the standard number in singular homology, is defined; the condition is that this number, applied to the intersection of the level sets ofImw with the hypersurfaceRew=0, vanish identically. In a CR structure, and in top dimension, this is shown to be equivalent to the property that the Levi form not be definite at any point—a property, that is likely to be also sufficient for local solvability.

     

Topological algebras with maximal regular ideals closed

It is shown that all maximal regular ideals in a Hausdorff topological algebra A are closed if the von Neumann bornology of A has a pseudo-basis which consists of idempotent and completant absolutely pseudoconvex sets. Moreover, all ideals in a unital commutative sequentially Mackey complete Hausdorff topological algebra A with jointly continuous multiplication and bounded elements are closed if the von Neumann bornology of A is idempotently pseudoconvex.     

On Fully Idempotent Modules

A submodule N of a module M is idempotent if N = Hom(M, N)N. The module M is fully idempotent if every submodule of M is idempotent. We prove that over a commutative ring, cyclic idempotent submodules of any module are direct summands. Counterexamples are given to show that this result is not true in general. It is shown that over commutative Noetherian rings, the fully idempotent modules are precisely the semisimple modules. We also show that the commutative rings over which every module is fully idempotent are exactly the semisimple rings. Idempotent submodules of free modules are characterized.     

An equational theory for trilattices

This paper presents the new algebra of trilattices, which are understood as the triadic generalization of lattices. As with lattices, there is an order-theoretic and an algebraic approach to trilattices. Order-theoretically, a trilattice is defined as a triordered set in which six triadic operations of some small arity exist. The Reduction Theorem guarantees that then also all finitary operations exist in trilattices. Algebraically, trilattices can be characterized by nine types of trilattice equations. Apart from the idempotent, associative, and commutative laws, further types of identities are needed such as bounds and limits laws, antiordinal, absorption, and separation laws. The similarities and differences between ordered and triordered sets, lattices and trilattices are discussed and illustrated by examples. Received May 26, 1998; accepted in final form May 7, 1999.     

Idempotent Totally Symmetric Operations on Finite Posets

An n-ary operation f is totally symmetric if it obeys the identity f(x ₁,...,x _n )=f(y ₁,...,y _n ) for all sets of variables such that {x ₁,...,x _n }={y ₁,...,y _n }. We characterize finite posets admitting an n-ary idempotent totally symmetric operation for all n. The characterization is expressed in terms of zigzags, special objects related to the poset. Some open problems concerning idempotent Malcev conditions for order primal algebras are mentioned.     

E-inversive semigroups with a completely simple kernel

We show that an E-inversive semigroup S has a completely simple kernel K_S if and only if it contains a primitive idempotent (in that case, K_S is the set-theoretic union of the groups eSe, where e is a primitive idempotent of S). Along the way, some equivalent conditions for a semigroup to be E-inversive are given. Moreover, some applications of the above theorem will be pointed out.     

On the Asymptotic Solution of Non-Hamiltonian Systems Exhibiting Sustained Resonance

With the exception of some special examples, much of the literature on the formal construction of asymptotic solutions of systems exhibiting sustained resonance concerns Hamiltonian problems, for which the reduced problem is of order two when a single resonance is present. In the Hamiltonian case, the resonance manifold is a curve that is explicitly defined by the governing equations and is independent of the actual sustained resonance solution. When the basic standard form system is non-Hamiltonian, with M slow and N fast variables, the corresponding reduced problem is of order M + 1; in general it involves all of the slow variables, P₁,…, P_M, plus the resonant phase Q. In this paper, the solution of a general non-Hamiltonian system in standard form is formally constructed for the case of a single sustained resonance. First, a well-known example is reviewed, for which the projection of the solutions on the resonance manifold can be derived a priori, independent of the evolution of Q. Then, the general case is solved, using a generalization of the multiple scale method of Kuzmak-Luke, where knowledge of the asymptotic solution for Q (as well as higher-order terms) is needed to define the projection of the solution on the resonance manifold. The results simplify significantly when initial conditions are chosen exactly on the resonance manifold; the modifications necessary for arbitrary initial conditions are also given. Two examples are discussed in detail to illustrate the procedure. The asymptotic results are confirmed for several test cases by comparison with numerical integrations of the exact equations.     

Sets of Periods for Expanding Maps on Flat Manifolds

 It is proven that the sets of periods for expanding maps on n-dimensional flat manifolds are uniformly cofinite, i.e. there is a positive integer m ₀, which depends only on n, such that for any integer , for any n-dimensional flat manifold ℳ and for any expanding map F on ℳ, there exists a periodic point of F whose least period is exactly m. (Received 10 April 1998; in revised form 20 January 1999)     

Optimal control under persistent disturbances

Let f be an orientation-preserving Morse-Smale diffeomorphism of an n-dimensional (n ≥ 3) closed orientable manifold M ⁿ . We show the possibility of representing the dynamics of f in a “source-sink” form. The roles of the “source” and “sink” are played by invariant closed sets one of which, A _f , is an attractor, and the other, R _f , is a repeller. Such a representation reveals new topological invariants that describe the embedding (possibly, wild) of stable and unstable manifolds of saddle periodic points in the ambient manifold. These invariants have allowed us to obtain a classification of substantial classes of Morse-Smale diffeomorphisms on 3-manifolds. In this paper, for any n ≥ 3, we describe the topological structure of the sets A _f and R _f and of the space of orbits that belong to the set M ⁿ \ (A _f ∪ R _f ).     

Idempotent structure of Clifford algebras

Spinor spaces can be represented as minimal left ideals of Clifford algebras and they are generated by primitive idempotents. Primitive idempotents of the Clifford algebras R _{p, q} are shown to be products of mutually nonannihilating commuting idempotent % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabaGaaiaacaqabeaadaqaaqGaaO% qaamaaleaaleaacaaIXaaabaGaaGOmaaaaaaa!3DBD!\[{\textstyle{1 \over 2}}\]2}}\](1+e _T ), where the k=q–r _q–p basis elements e _T satisfy e _T ²=1. The lattice generated by a set of mutually annihilating primitive idempotents is examined. The final result characterizes all Clifford algebras R _{p, q} with an anti-involution such that each symmetric elements is either a nilpotent or then some right multiple of it is a nonzero symmetric idempotent. This happens when p+q<-3 and (p, q)(2, 1).     

On some classes of regular semigroups

An idempotent e of a semigroup S is called right [left] principal (B.R. Srinivasan, [2]) if fef=fe [fef=ef] for every idempotent f of S. Say that S has property (LR) [(LR₁)] if every ℒ-class of S contains atleast [exactly] one right principal idempotent. There and six further properties obtained by replacing, ‘ℒ-class’ by ‘ℛ-class’ and/or ‘right principal’ by ‘left principal’ are examined. If S has (LR₁), the set of right principal elementsa of S (aa′ is right principal for some inversea′ ofa) is an inverse subsemigroup of S, generalizing a theorem of Srinivasan [2] for weakly inverse semigroups. It is shown that the direct sum of all dual Schützenberger representations of an (LR) semigroup is faithful (cf[1], Theorem 3.21, p. 119). Finally, necessary and sufficient conditions are given on a regular subsemigroup S of the full transformation semigroup on a set in order that S has each of the properties (LR), (LR₁), etc.