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The paper studies the orthogonality relation graphs for noncommutative rings. Known results on the diameters of the connected components of simple Artinian rings are generalized to larger ring classes, particularly, to semisimple Artinian and semiprime two-sided Goldie rings. Also the presence of isolated vertices in the graphs of the above ring classes is considered, and the behavior of the diameter function under taking a direct sum for pairs of arbitrary rings is studied. [ABSTRACT FROM AUTHOR]

Upper and lower estimates for the maximal commutator length of a noncentral element of the elementary subgroup of the general linear group over a skew-field are proved on the basis the maximal commutator length of an element of the multiplicative group of this skew-field. [ABSTRACT FROM AUTHOR]

This review paper is a continuation of two previous review papers devoted to properties of modules over discrete valuation domains. The first part of this work was published in the volumes 121 (Part I, Chaps. 1–4, Secs. 1–22) and 122 (Part II, Chaps. 5–8, Secs. 23–39) of the series Itogi Nauki i Tekhniki. Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory (English translation: Journal of Mathematical Sciences (New York), vol. 145, No. 4, pp. 4997–5117 (2007), and vol. 151, No. 5, pp. 3225–3371 (2008)). In this review paper, we preserve the numeration of chapters and sections of parts I and II. The first part consists of Chapter 9 "Appendix," Secs. 40–42. In Sec. 40, we consider p-adic torsion-free modules with isomorphic automorphism groups. Section 41 is devoted to torsion-free modules over a complete discrete valuation domain with isomorphic radicals of their endomorphism rings. The volume of the paper does not allow us to provide the proofs of all the results that appeared after publication of the previous parts and directly related to the issues under consideration in it. In the final Sec. 42, we describe some of these new results. [ABSTRACT FROM AUTHOR]

The word maps w ˜ : GL m D 2 k → GL n D and w ˜ : D ∗ 2 k → D ∗ for a word w = ∏ i = 1 k x i y i , where D is a division algebra over a field K, are considered. It is proved that if w ˜ D ∗ 2 k = D ∗ D ∗ , then w ˜ GL n D ⊃ E n D \ Z E n D , where En(D) is the subgroup of GLn(D), generated by transvections, and Z(En(D)) is its center. Furthermore if, in addition, n > 2, then w ˜ E n D ⊃ E n D \ Z E n D . The proof of the result is based on an analog of the "Gauss decomposition with prescribed semisimple part" (introduced and studied in two papers of the second author with collaborators) in the case of the group GLn(D), which is also considered in the present paper. [ABSTRACT FROM AUTHOR]

The paper is devoted to studying the orthogonality graph of the matrix ring over a skew field. It is shown that for n ≥ 3 and an arbitrary skew field 픻, the orthogonality graph of the ring Mn(픻) of n × n matrices over a skew field 픻 is connected and has diameter 4. If n = 2, then the graph of the ring Mn(픻) is a disjoint union of connected components of diameters 1 and 2. As a corollary, the corresponding results on the orthogonality graphs of simple Artinian rings are obtained. [ABSTRACT FROM AUTHOR]

This paper is devoted to some applications of local algebras in geometry. We recall some properties of free finite-dimensional modules over local algebras of a certain type, so-called A-spaces in the sense of Macdonald, where A denotes the algebra considered. Using these properties, we study bilinear, special symmetric, and symplectic forms on A-spaces and obtain some their invariants. Properties of these spaces are used in the study of projective Klingenberg spaces over the ring A. We present fundamental notions of points and subspaces of projective Klingenberg spaces. We examine the neighborship property of points and homologies. We obtain a criterion of projective equivalence of quadrics on these spaces. [ABSTRACT FROM AUTHOR]

We use U. Brehm’s ideas to study cross-ratios, harmonic relations, and harmonic maps. For torsion-free modules over noncommutative principal ideal domains, von Staudt–Hua’s theorem is proved. Moreover, more general (nonbijective) harmonic maps with the classical definition of harmonic quadruples are calculated. [ABSTRACT FROM AUTHOR]

We discuss algebraic representations of mappings preserving arbitrary joins between submodule lattices. For a given join-preserving mapping $$ \bar{g}:\mathfrak{L}\left( {_RM} \right)\to \mathfrak{L}\left( {_SN} \right) $$ between submodule lattices, a representation is an R-balanced mapping h : B × M → N, where B is a bimodule such that $$ \left\langle {h\left( {B\times U} \right)} \right\rangle =\bar{g}(U) $$ for all $$ U\in \mathfrak{L}\left( {_RM} \right) $$. We begin by posing the question in a general abstract context and by defining the canonical subrepresentation, which is a representation if and only if there exists a representation. The problem is to give easy and natural conditions for the existence of a representation. We consider a very general situation for the mappings and give sufficient criteria for the existence of a representation. We also consider lattice isomorphisms. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of three-dimensional generalized rotations. We obtain relations between the parameters of the spinor representation of the group of three-dimensional generalized rotations and the coordinates of the initial and terminal points of rotation. Simple relations between elements of a three-dimensional orthogonal matrix of the basic representation and the Euler angles, on the one hand, and the coordinates of the initial and terminal points of rotation, on the other hand, were derived. The spinor method of solution of the inverse kinematic problem for spatial mechanisms with spherical pairs is developed and the corresponding algorithm is proposed. The obtained results allow one to reduce the three-dimensional problem of spatial motion control to the one-dimensional problem. Simple adaptive algorithms are suggested, by means of which various partial problems on the terminal control are solved under various terminal conditions. New algorithms of control of spatial rotations of manipulating robots are studied. [ABSTRACT FROM AUTHOR]

Relatively recent and new results on the enumeration of classes of projective congruent quadrics and classes of projective equivalent quadrics of projective spaces over a local ring R = 2 R with nilpotent principal maximal ideal are reflected in the paper. For the case of the basic ring R with nonprincipal maximal ideal, the enumerations of quadrics of projective plane are given up to projective equivalence. [ABSTRACT FROM AUTHOR]

In this paper, we bring together design theory and projective geometry over a ring. Both disciplines are well established, but the results on the interaction between them are rare and are scattered over the literature. Our main goal is to present the basics from either side, to develop, or at least sketch, the principal connections between them, and to make recommendations for further reading. [ABSTRACT FROM AUTHOR]

The aim of this work is to investigate the validity of the fundamental theorem of affine geometry over rings with an invariant basis number. [ABSTRACT FROM AUTHOR]

In this paper, we obtain an upper estimate for the cardinality of a subset of the discrete torus over a field of three elements of which any four points do not form a nonsingular parallelogram. [ABSTRACT FROM AUTHOR]

A generalization of the theory of representations of perspective mappings by (semi)linear functions to rings is proposed. [ABSTRACT FROM AUTHOR]

The article presents the definition of finite geometries. At the end of the nineteenth century, it notes that finite geometries appear as a synthesis of combinatorics, group-theoretic ideas, and statistical data-processing methodology. It is also usually considered in terms of incidence structures or general combinatorial theory. The author mentions that geometric and combinatorial tools has proved to be fruitful for the development of geometrical number theory and graph theory.

We show that under certain weak conditions on the module R M, every mapping between the submodule lattices which preserves arbitrary joins and “disjointness” has a unique representation of the form f( u) = 〈 h[ S B R × R U]〉 for all u ∈ , where S B R is some bimodule and h is an R-balanced mapping. Furthermore, f is a lattice homomorphism if and only if B R is flat and the induced S-module homomorphism is monic. If S N also satisfies the same weak conditions, then f is a lattice isomorphism if and only if B R is a finitely generated projective generator, S ≅ End( B R ) canonically, and is an S-module isomorphism, i.e., every lattice isomorphism is induced by a Morita equivalence between R and S and a module isomorphism. [ABSTRACT FROM AUTHOR]

To obtain the representation ( L, R) of Lie algebras over the ring Λ, we construct the lattice of subrepresentations ℒ( L, R). Relations between the algebras L and R and the lattice ℒ( L, R) are studied. It turns out that in some cases the isomorphism of the lattice ℒ( L, R) can be continued so as to obtain a wider sublattice ℒ( LλR) consisting of subalgebras of a semidirect product LλR. [ABSTRACT FROM AUTHOR]

We present a solution of the problem of construction of a normal diagonal form for quadratic forms over a local principal ideal ring R = 2 R with a QF-scheme of order 2. We give a combinatorial representation for the number of classes of projective congruence quadrics of the projective space over R with nilpotent maximal ideal. For the projective planes, the enumeration of quadrics up to projective equivalence is given; we also consider the projective planes over rings with nonprincipal maximal ideal. We consider the normal form of quadratic forms over the field of p-adic numbers. The corresponding QF-schemes have order 4 or 8. Some open problems for QF-schemes are mentioned. The distinguished finite QF-schemes of local and elementary types (of arbitrarily large order) are realized as the QF-schemes of a field. [ABSTRACT FROM AUTHOR]

In the second part of the paper, we study torsion-free modules and mixed modules. We analyze the possibility of the isomorphism of two modules with isomorphic endomorphism rings. We touch on several questions about transitive and fully transitive modules. [ABSTRACT FROM AUTHOR]

The well-known problem of classical mechanics considered by Bertrand (1857) and Darboux (1901) is reviewed in the context of Cartan’s geometry. [ABSTRACT FROM AUTHOR]

Let G = G(Φ, K) be a Chevalley group over a field K of characteristic ≠ 2. In the present paper, we classify the subgroups of G generated by triples of long root subgroups, two of which are opposite, up to conjugacy. For finite fields, this result is contained in papers by B. Cooperstein on the geometry of root subgroups, whereas for SL (n, K) it is proved in a paper by L. Di Martino and the first-named author. All interesting items of our list appear in deep geometric results on abstract root subgroups and quadratic actions by F. Timmesfeld and A. Steinbach, and also by E. Bashkirov. However, for applications to the groups of type E l, we need a detailed justification of this list, which we could not extract from the published papers. That is why in the present paper, we produce a direct elementary proof based on the reduction to D 4, where the question is settled by straightforward matrix calculations. Bibliography: 73 titles. [ABSTRACT FROM AUTHOR]

The article presents several topics related to discrete valuation domains. These include the concepts of prime elements, endomorphism rings, torsion-free modules, the problem of existence of isomorphism, and transitive modules as well as the fundamentals of the theory of modules over discrete valuation domains. Detailed discussions about these topics are offered.

In the present paper, we characterize ⋀n( GL(n, R)) over any commutative ring R as the connected component of the stabilizer of the Plücker ideal. This folk theorem is classically known for algebraically closed fields and should also be well known in general. However, we are not aware of any obvious reference, so we produce a detailed proof, which follows a general scheme developed by W.C.Waterhouse. The present paper is a technical preliminary to a subsequent paper, where we construct the decomposition of transvections in polyvector representations of GL n. Bibliography: 50 titles. [ABSTRACT FROM AUTHOR]

The present paper is devoted to a detailed computer study of the action of the Chevalley group G( E 6, R) on the minimal module V( ῶ 1). Our main objectives are an explicit choice and a tabulation of the signs of structure constants for this action, compatible with the choice of a positive Chevalley base, the construction of multilinear invariants and equations on the matrix entries of matrices from G( E 6, R) in this representation, and an explicit tabulation of root elements. Bibliography: 170 titles. [ABSTRACT FROM AUTHOR]

We construct a notion of a generalized geometrical lattice, allowing us to obtain an analog of matroid theory for the case of arbitrary principal ideal rings. [ABSTRACT FROM AUTHOR]

The article discusses the process of understanding the geometry and algebra of the classical three-web theory with foliations. It provides a definition of three-web under the manifold of three generic foliations. It also explores the classical theory of the web, while considering its accuracy based on local diffeomorphisms and isotopic transformations.

This paper is a survey of results on symmetries of crystals and of some results on a mathematical approach to the theory of quasicrystals and their symmetries. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the validity of the fundamental theorem of affine geometry over rings with an invariant basis number. [ABSTRACT FROM AUTHOR]

The interrelations between the theory of linear transformations preserving matrix invariants and different branches of mathematics are surveyed here. The preferences are given for those methods and motivations to study these transformations that arise from general algebra. [ABSTRACT FROM AUTHOR]

In this survey, we consider the papers devoted to elementary properties of groups with the main focus on linear groups. We also consider problems on groups that are not linear but not in detail. We do not pretend to aim for a complete treatment of the problem, but we hope that the main results of the subject will be considered. In the part on results obtained before 1984, we are guided by the second chapter of the survey of V. N. Remeslennikov and V. A. Roman'kov. The first section of the survey contains basic necessary information about model theory: definitions of model, first-order language, elementary and universal theories, axiomatizability and unaxiomatizability, saturation, elementary equivalence, and ultraproducts. We also present some main theorems concerned with the notions of stability, axiomatizability, saturation, ultrafilters, and ultraproducts. The second section is devoted to the classification of groups up to their elementary properties. We present the theorem W. Szmielewa theorem on the elementary equivalence of Abelian groups, some theorems of A. G. Myasnikov and V. N. Remeslennikov on nilpotent Q-groups of finite rank, and some results of F. Oger concerned with elementary equivalence of different (finitely generated, mainly finite-by- Abelian or finite-by-nilpotent) groups. Section 3 is the main one in the survey. It contains results of elementary classification of classical linear groups: the theorem of A. I. Maltsev on the elementary equivalence of groups GL, PGL, SL, and PSL over fields, results of C. I. Beidar and A. V. Mikhalev generalizing the previous theorem to the case of groups GLn over associative rings and skew-fields, the papers of E. I. Bunina on the elementary equivalence of unitary linear groups over fields, skew-fields, and rings and Chevalley groups over fields, and the results of V. Tolstykh on the elementary equivalence of infinite-dimensional linear groups over skew-fields. There are some other results on the elementary equivalence of groups and related structures, which are included in Sec. 4. These are results on partition lattices, lattices of subalgebras of free algebras, modules and their tensor products, filtered and wreath products of groups, and some other results. The fifth section is devoted to categoricity and stability. We describe some results on co-categorical and co-stable groups, ωi-categorical groups (including theorems of B. I. Zilber on ω1-categoricity of simple algebraic groups over algebraically closed fields). In the last, sixth section, we consider the main results on the decidability and undecidability of various group theories, including basic methods for proving the decidability and undecidability, and also results on universal group theories. [ABSTRACT FROM AUTHOR]

The subgroups of the split orthogonal group Γ = GO(n,R) of degree n over a commutative ring R with 2 ∈ R* that contain the elementary subgroup of a regularly embedded semisimple group F are described. It is shown that if the ranks of all irreducible components of F are at least 4, then the description of its overgroups is standard in the sense that for any intermediate subgroup H there exists a unique orthogonal net of ideals such that H lies between the corresponding net subgroup and its normalizer in Γ. A similar result for subgroups of the general linear group GL(n,R) with irreducible components of ranks at least 2 was obtained by Z. I. Borewicz and the present author. Examples which show that if F has irreducible components of ranks 2 or 3, then the standard description does not hold were constructed. The paper is based on the previous publications by the author, where similar results were obtained in some special cases, but the proof is based on a new computational trick. Bibliography: 76 titles. [ABSTRACT FROM AUTHOR]

We study the branching of representations of a p-elementary quadratic form by a genus of positive definite locally p-two-dimensional forms. A primitive representation of a p-elementary form is decomposed into a direct sum of minimal indecomposable representations; the latter representations are found in an explicit form. For the case of branching, we find local multipliers of the weight of representations of a form by a genus. As an application, we calculate the number of embeddings into the classical root lattices. The method of orthogonal complement is applied in constructing new genera of quadratic forms. Bibliography: 9 titles. [ABSTRACT FROM AUTHOR]

In the first paper of the present series, we proved the standardness of subgroups containing a split maximal torus in the split orthogonal group SO( n,R) over a commutative semilocal ring R for the following two situations: (1) n is even; (2) n is odd and R = K is a field. In the present paper, we prove the standardness of intermediate subgroups over a semilocal ring R in the case of an odd n. Together with the preceding papers by Z. I. Borewicz, the author, and E. V. Dybkova, this paper completes the description of the overgroups of split maximal tori in the classical groups over semilocal rings. The analysis of odd orthogonal groups turned out to be technically much more difficult than other classical cases. Unlike all preceding papers, the proof of the key step in the reduction to the case of a field relies on calculations in a class of semisimple elements that are neither microweight elements nor semisimple root elements. Bibliography: 55 titles. [ABSTRACT FROM AUTHOR]

Smooth manifolds with degenerate symmetric bilinear form of constant signature on tangent spaces are studied. Such manifolds are used in physical theories of unified interactions including gravitation. Necessary and sufficient conditions for the existence of a Levi-Civita connection on such a manifold are given. Bibliography: 6 titles. [ABSTRACT FROM AUTHOR]

Classical geodesic embeddings in the Grassmann manifold $$G_p^+(V)\subset\Lambda(V)$$ are described algebraically by means of the exterior algebra Λ( V). The group I0( G+) of isometries of the Grassmannian is described without the theory of Lie groups and algebras. Interior and exterior properties of Grassmann manifolds are proved with the help of the equivariant Plücker embedding. The isomorphism type of the group of twists of the Grassmann manifold around an arbitrary geodesic is described. Bibliography: 16 titles. [ABSTRACT FROM AUTHOR]

Let D be a division algebra with a fixed involution, and let V be the corresponding unitary space over D with T-condition (see N. Bourbaki, Algebre). For a pair of totally isotropic subspaces u,v [less than or equal to] V, the double cosets [P.sub.u] [gamma] [P.sub.v] of their stabilizers [P.sub.u], [P.sub.v] in [TAU] = SU(V) are considered. A description of the cosets [P.sub.u] [gamma] [P.sub.v] in terms of the intersection distasnce [d.sub.in](u, [gamma](v)) and the Witt index of u + [gamma](v) is given. Bibliography: 9 titles., UDC 512.74 INTRODUCTION In [8], the notion of linear Kleiman group was introduced: a linear group G [less than or equal to] GL(V) is called a Kleiman group if for [...]

This paper is devoted to some applications of local algebras in geometry. We recall some properties of free finite-dimensional modules over local algebras of a certain type, so-called A-spaces in the sense of Macdonald, where A denotes the algebra considered. Using these properties, we study bilinear, special symmetric, and symplectic forms on A-spaces and obtain some their invariants. Properties of these spaces are used in the study of projective Klingenberg spaces over the ring A. We present fundamental notions of points and subspaces of projective Klingenberg spaces. We examine the neighborship property of points and homologies. We obtain a criterion of projective equivalence of quadrics on these spaces., CONTENTS 1. A-Spaces and Their Homomorphisms 2. λ-Bilinear Forms on A-Spaces 3. Projective Klingenberg Spaces References UDC 514.7 This paper is devoted to some applications of local algebras in geometry. [...]

Chapter 9. In the limit In this chapter, we start discussing the behavior of the group GL (n, R) in the case where n is large with respect to some [...]

In this paper, we obtain an upper estimate for the cardinality of a subset of the discrete torus over a field of three elements of which any four points do not form a nonsingular parallelogram., UDC 511.29 1. Basic Definitions Let G be a group, written additively. A subset A of G is called a difference set if, for any x, y, u, v [member [...]