Universal gradings of orders

For commutative rings, we introduce the notion of a universal grading, which can be viewed as the “largest possible grading”. While not every commutative ring (or order) has a universal grading, we prove that every reduced order has a universal grading, and this grading is by a finite group. Examples of graded orders are provided by group rings of finite abelian groups over rings of integers in number fields. We also generalize known properties of nilpotents, idempotents, and roots of unity in such group rings to the case of graded orders; this has applications to cryptography. Lattices play an important role in this paper; a novel aspect is that our proofs use that the additive group of any reduced order can in a natural way be equipped with a lattice structure.     

Planar Functions and Planes of Lenz-Barlotti Class II

Planar functions were introduced by Dembowski and Ostrom [4] to describe projective planes possessing a collineation group with particular properties. Several classes of planar functions over a finite field are described, including a class whose associated affine planes are not translation planes or dual translation planes. This resolves in the negative a question posed in [4]. These planar functions define at least one such affine plane of order 3^e for every e 4 and their projective closures are of Lenz-Barlotti type II. All previously known planes of type II are obtained by derivation or lifting. At least when e is odd, the planes described here cannot be obtained in this manner.     

Collineation groups irreducible on the components of a translation plane

We investigate finite translation planes of odd dimension over their kernels in which the translation complement induces on each component l a permutation group whose order is divisible by a p-primitive divisor. Using results of this investigation, we show that rank 3 affine planes of odd dimension over their kernels are either generalized André planes or semi-field planes. A similar result is given for translation planes having a collineation group which is doubly transitive on each affine line; besides the above two possibilities, there is a third possibility; the plane has order 27, the translation complement is doubly transitive on , and SL(2, 13) is contained in the translation complement.We also consider translation planes of odd dimension over their kernels which have a collineation group isomorphic to SL(2, w) with w prime to 5 and the characteristic, and having no affine perspectivity. We show that such planes have order 27, the prime power w=13, and the given group together with the translations forms a doubly transitive collineation group on {ie153-1}. This indicates quite strongly that the Hering translation plane of order 27 is unique with respect to the above properties.Both authors supported in part by NSF Grant No. MCS76-0661 A01.     

The Cohen Macaulay Property for Noncommutative Rings

Let R be a noetherian ring which is a finite module over its centre Z(R). This paper studies the consequences for R of the hypothesis that it is a maximal Cohen Macaulay Z(R)-module. A number of new results are proved, for example projectivity over regular commutative subrings and the direct sum decomposition into equicodimensional rings in the affine case, and old results are corrected or improved. The additional hypothesis of homological grade symmetry is proposed as the appropriate extra lever needed to extend the classical commutative homological hierarchy to this setting, and results are proved in support of this proposal. Some speculations are made in the final section about how to extend the definition of the Cohen-Macaulay property beyond those rings which are finite over their centres.     

Infinite-dimensional diagonalization and semisimplicity

We characterize the diagonalizable subalgebras of End(V), the full ring of linear operators on a vector space V over a field, in a manner that directly generalizes the classical theory of diagonalizable algebras of operators on a finite-dimensional vector space. Our characterizations are formulated in terms of a natural topology (the “finite topology”) on End(V), which reduces to the discrete topology in the case where V is finite-dimensional. We further investigate when two subalgebras of operators can and cannot be simultaneously diagonalized, as well as the closure of the set of diagonalizable operators within End(V). Motivated by the classical link between diagonalizability and semisimplicity, we also give an infinite-dimensional generalization of the Wedderburn–Artin theorem, providing a number of equivalent characterizations of left pseudocompact, Jacoboson semisimple rings that parallel various characterizations of artinian semisimple rings. This theorem unifies a number of related results in the literature, including the structure of linearly compact, Jacobson semsimple rings and cosemisimple coalgebras over a field.     

Maximal integral point sets in affine planes over finite fields

Motivated by integral point sets in the Euclidean plane, we consider integral point sets in affine planes over finite fields. An integral point set is a set of points in the affine plane over a finite field F_q, where the formally defined squared Euclidean distance of every pair of points is a square in F_q. It turns out that integral point sets over F_q can also be characterized as affine point sets determining certain prescribed directions, which gives a relation to the work of Blokhuis. Furthermore, in one important sub-case, integral point sets can be restated as cliques in Paley graphs of square order.In this article we give new results on the automorphisms of integral point sets and classify maximal integral point sets over F_q for q≤47. Furthermore, we give two series of maximal integral point sets and prove their maximality.     

An axiomatic look at the Erdős-Trost problem

The Erd?s-Trost problem can be formulated in the following way: “If the triangle XY Z is inscribed in the triangle ABC—with X, Y, and Z on the sides BC, CA, and AB, respectively—then one of the areas of the triangles BXZ, CXY , AY Z is less than or equal to the area of the triangle XY Z.” There are many different solutions for this problem. In this note we take up a very elementary proof (due to Szekeres) and deduce that the class of ordered translation planes is the level in the hierarchy of affine planes where the Erd?s-Trost statement still holds true. We also look at the conditions an absolute plane needs to satisfy for the validity of the Erd?s-Trost statement.     

Sard theorems for Lipschitz functions and applications in optimization

We establish a “preparatory Sard theorem” for smooth functions with a partially affine structure. By means of this result, we improve a previous result of Rifford [17, 19] concerning the generalized (Clarke) critical values of Lipschitz functions defined as minima of smooth functions. We also establish a nonsmooth Sard theorem for the class of Lipschitz functions from R^d to R^p that can be expressed as finite selections of C^k functions (more generally, continuous selections over a compact countable set). This recovers readily the classical Sard theorem and extends a previous result of Barbet–Daniilidis–Dambrine [1] to the case p > 1. Applications in semi-infinite and Pareto optimization are given.     

Formal Hecke algebras and algebraic oriented cohomology theories

In the present paper, we generalize the construction of the nil Hecke ring of Kostant–Kumar to the context of an arbitrary formal group law, in particular, to an arbitrary algebraic oriented cohomology theory of Levine–Morel and Panin–Smirnov (e.g., to Chow groups, Grothendieck’s \(K_0\) , connective \(K\) -theory, elliptic cohomology, and algebraic cobordism). The resulting object, which we call a formal (affine) Demazure algebra, is parameterized by a one-dimensional commutative formal group law and has the following important property: specialization to the additive and multiplicative periodic formal group laws yields completions of the nil Hecke and the 0-Hecke rings, respectively. We also introduce a formal (affine) Hecke algebra. We show that the specialization of the formal (affine) Hecke algebra to the additive and multiplicative periodic formal group laws gives completions of the degenerate (affine) Hecke algebra and the usual (affine) Hecke algebra, respectively. We show that all formal affine Demazure algebras (and all formal affine Hecke algebras) become isomorphic over certain coefficient rings, proving an analogue of a result of Lusztig.     

On the shape of solution sets of systems of (functional) equations

Solution sets of systems of linear equations over fields are characterized as being affine subspaces. But what can we say about the “shape” of the set of all solutions of other systems of equations? We study solution sets over arbitrary algebraic structures, and we give a necessary condition for a set of n-tuples to be the set of solutions of a system of equations in n unknowns over a given algebra. In the case of Boolean equations we obtain a complete characterization, and we also characterize solution sets of systems of Boolean functional equations.     

Kennzeichnungen der affinen Bilder von Körpern konstanter Breite

Characterizing ellipsoids by plane shadow boundaries BLASCHKE also showed that ellipsoids are characterized by the fact that their plane projections are bounded by Radon curves (R-curves). It follows that the affine images of convex bodies of constant width (here called “affine Gleichdicke”) are characterized by having projections which are bounded by P-curves (the non-symmetrical analogues of R-curves). We thereby obtain two further characteristic properties of “affine Gleichdicke”:

1. The vertices of the largest inscribed cross polytope Q lie on the faces of the smallest circumscribed parallelotope P.
2. Their volumes are related by ¦P¦=¦Q¦·n!.

     

Affine semigroups acting properly discontinuously on a hyperbolic space

In this paper we study the dynamics of properly discontinuous and crystallographic affine semigroups leaving a hyperbolic form, i.e. a quadratic from of signature (n, 1) invariant. The motivating question here is a question stated by H. Abels, G. Margulis and the author: Is the Zariski closure of a crystallographic affine semigroup leaving a hyperbolic form invariant a virtually solvable group? We proved that that for n = 2 the answer is “yes”.     

Über lokalarchimedische Anordnungen projektiver Ebenen

The question, whether the Archimedean ordering of only one of the ternary rings of a projective plane implies that is Archimedean, i.e. that every ternary ring of is Archimedean, is answered in the negative by the construction of local-Archimedean orderings of free planes. There exists even Archimedean affine planes with non-Archimedean associated projective planes.     

Affine Barbilian structures

W.Leissner has characterized, by geometric axioms, the affine Barbilian planes over a Z-ring (i.e, a ring with 1 such that ab=1 ba=1) [10].The aim of the present paper is to characterize correspondingly the affine Barbilian planes over an arbitrary ring with 1. First we shall deal with the translation Barbilian planes, which generalize Leissner's parallelodromic planes [11]. The paper concludes with a study of the kernel of the translation Barbilian plane.Here, the terms affine Barbilian structure and affine Barbilian plane are used in a more general sense than in [10] and [11]. Also, the definitions of translation and parallelodromy are slightly different from those in [10] and [11], insomuch that the invariance of the non-neighbour relation is not postulated any more, this being a consequence in a translation Barbilian plane.In H.J.Arnold's geometry of rings, for any two distinct points, there exists a smallest line incident with them [1]. This property, assumed only for the non-neighbour pairs of points, will replace the usual postulate that two non-neighbour points are incident with exactly one line. Thus, ideas of D.Barbilian [2] and H. J.Arnold [1] are combined with methods of affine ring-geometry due to J.Hjelmslev [5], [6], W.Klingenberg [7], [8], [9], H.Lüneburg [12], W.Benz [3], [4], W.Leissner [10], [11], and others. Many parts of the proofs in [10] and [11] could be used here almost unchanged, under relaxed assumptions.     

Unimodular graphs and Eisenstein sums

Motivated in part by combinatorial applications to certain sum-product phenomena, we introduce unimodular graphs over finite fields and, more generally, over finite valuation rings. We compute the spectrum of the unimodular graphs, by using Eisenstein sums associated with unramified extensions of such rings. We derive an estimate for the number of solutions to the restricted dot product equation \(a\cdot b=r\) over a finite valuation ring. Furthermore, our spectral analysis leads to the exact value of the isoperimetric constant for half of the unimodular graphs. We also compute the spectrum of Platonic graphs over finite valuation rings, and products of such rings—e.g., \(\mathbb {Z}/(N)\). In particular, we deduce an improved lower bound for the isoperimetric constant of the Platonic graph over \(\mathbb {Z}/(N)\).     

Konstruktion topologischer affiner Ebenen mit nichtstetigem Parallelismus durch Knicken von Geraden

We prove the existence of a class of topological affine planes having non-continuous parallelism by using [2, Satz 5.2]. For this, we introduce a new method of constructing affine Salzmann-planes with a monotonically increasing slope (see 2.1) by bending lines on two special curves, which are not necessary lines. Furthermore, the limit inferior of a sequence of topological planes with fixed point space is defined. As application of our new method, we construct a sequence of affine Salzmann-planes such that the limit inferior of this sequence is again an affine Salzmann-plane and fulfils the assumptions of [2, Satz 5.2]. Applying this theorem repeatedly, we get a sequence of non-isomorphic topological affine subplanes with non-continuous parallelism.     

A characterization of special laguerre planes and extended dual affine planes

A special Laguerre plane is a nondegenerate transversal 3-design such that the residue of each point is a dual affine plane. A special Laguerre plane is equivalent to an optimal code with three information digits and maximal length. An extended dual affine plane is an incidence structure (whose objects will be called points and blocks) such that the residue of each point is a dual affine plane, and each pair of points is in at least one block. Finite extended dual affine planes exist only of order 2, 4, and (dubiously) 10. We show that any finite incidence structure having the residue of each point a dual affine plane either is a transversal 3-design or has a block through each pair of points. Hence theorem: If a finite nondegenerate connected incidence structure has the residue of each point a dual affine plane, then is either an extended dual affine plane or a special Laguerre plane. This research was partially supported by NSF Grant MCS-8102361.     

Steiner triple systems with doubly transitive automorphism groups: A corollary to the classification theorem for finite simple groups

Assuming that the classification theorem for finite simple groups is complete, a conjecture of M. Hall (Proc. Sympos. Pure Math.6 (1962), 47–66; and in“Proceedings of the International Conference on Theory of Groups”, pp. 115–144, Australian National University, Canberra, Australia, 1965) that a Steiner triple system with a doubly transitive automorphism group is a projective or affine geometry, is verified.     

On Strongly Clean Matrix and Triangular Matrix Rings

A ring R with identity is called “clean” if every element of R is the sum of an idempotent and a unit, and R is called “strongly clean” if every element of R is the sum of an idempotent and a unit that commute. Strongly clean rings are “additive analogs” of strongly regular rings, where a ring R is strongly regular if every element of R is the product of an idempotent and a unit that commute. Strongly clean rings were introduced in Nicholson (1999 Nicholson , W. K. (1999). Strongly clean rings and Fitting's lemma. Comm. Algebra 27:3583–3592. [CSA] [Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) where their connection with strongly π-regular rings and hence to Fitting's Lemma were discussed. Local rings and strongly π-regular rings are all strongly clean. In this article, we identify new families of strongly clean rings through matrix rings and triangular matrix rings. For instance, it is proven that the 2 × 2 matrix ring over the ring of p-adic integers and the triangular matrix ring over a commutative semiperfect ring are all strongly clean.