4-Transitivity and 6-figures in some Moufang-Klingenberg planes

In this paper, the Moufang-Klingenberg plane over a local alternative ring R of dual numbers is studied. It is shown that its collineation group is transitive on quadrangles. It is therefore shown that the coordinatization of these Moufang-Klingenberg planes is independent of the choice of the coordinatization quadrangle. Also, the concept of 6-figures is extended to these Moufang-Klingenberg planes and it is shown that any 6-figure corresponds to only one inversible m ∈ R.     

Cross-ratios in Moufang-Klingenberg planes

Certain geometric groups operating on a line g in a Moufang-Klingenberg plane. are described algebraically in terms of the underlying alternative ring R. For the case of the dual numbers R=A+A (A alternative field, ²=0) a notion of cross-ratio is introduced on the line. We establish some connections between the geometric groups and the cross-ratio which are well known from classical projective planes.     

Noncommutative Jordan superalgebras of degree <Emphasis Type="Italic">n</Emphasis> > 2

We prove a coordinatization theorem for noncommutative Jordan superalgebras of degree n > 2, describing such algebras. It is shown that the symmetrized Jordan superalgebra for a simple finite-dimensional noncommutative Jordan superalgebra of characteristic 0 and degree n > 1 is simple. Modulo a “nodal” case, we classify central simple finite-dimensional noncommutative Jordan superalgebras of characteristic 0.     

Fractional dimensions in semifields of odd order

A finite semifield D is considered a fractional dimensional semifield if it contains a subsemifield E such that λ := log_|E||D| is not an integer. We develop spread-theoretic tools to determine when finite planes admit coordinatization by fractional semifields, and to find such semifields when they exist. We use our results to show that such semifields exist for prime powers 3 ⁿ whenever n is an odd integer divisible by 5 or 7.     

Projective remoteness planes

A new axiomatization involving incidence and remoteness of planes with nondivision coordinate rings is introduced and a coordinatization theorem is obtained. A geometric process of splitting points and lines to obtain another plane with the same coordinates is described. It is also shown that a group of Steinberg type is parametrized by a nonassociative ring. The notion of elementary basis sets for an associative ring is introduced and constructions of projective and affine planes are given. A plane with reflections determining a system of rotations is shown to have commutative, associative coordinates.     

Les triplets de la variété de Grassmann G2(R6)

It is shown that a triple of two-dimensional planes in R⁶ is determined up to isometry by nine angular invariants – the six critical angles and the three angles between the critical bases in each plane – and four signs. Conditions of existence of these triples are then analyzed. These results are used to study some noteworthy triples.     

R2-planes with 3-dimensional automorphism group fixing precisely a line

It is shown that the class of these R²-planes consists of those listed by STRAMBACH 1970, those added by OSTMANN 1975, and those added by GROH 1981. This last class consists of sums of two hyperbola planes. Furthermore, the isomorphism types are determined. All investigations are carried out for the much more general situation of an R²-plane with Δ-invariant screen, Δ ? R².     

Generators for complemented modular lattices and the von Neumann–Jónsson coordinatization theorems

Extending work of von Neumann, Jónsson has shown that each complemented modular lattice, L admitting a large partial n-frame with n ≥ 4, or with n ≥ 3 and L Arguesian, can be coordinatized as the lattice of all principal right ideals of some regular ring. His proof built on the embedding of L into the subgroup lattice of an abelian group which follows from Frink’s embedding of L into to a direct product of subspace lattices of irreducible projective spaces and coordinatization of the latter. We offer a proof which, in addition to these results, employs only some elementary linear algebra. Luca Giudici’s thesis [6] is an important source for this approach.     

Detecting Planar Patches in an Unorganised Set of Points in Space

Given n points in 3D, sampled from k original planes (with sampling errors), a new probabilistic method for detecting coplanar subsets of points in O(k ⁶) steps is introduced. The planes are reconstructed with small probability of error. The algorithm reduces the problem of reconstruction to the problem of clustering in R ³ and thereby produces effective results. The algorithm is significantly faster than other known algorithms in most cases.     

Preinjective modules and finite representation type of artinian rings

Let R be a left artinian ring such that every finitely presented right .ft-module is of finite endolength. It is shown that the cardinality of the set of isomorphism classes of preinjective right R-modules is less than or equal to the cardinality of the set of isomorphism classes of preprojective left R-modules, and R is of finite representation type if and only if these cardinal numbers are finite and equal to each other. As a consequence, we deduce a theorem, due to Herzog [17], asserting that a left pure semisimple ring R is of finite representation type if and only if the number of non-isomorphic preinjective right R-modules is the same as the number of non-isomorphic preprojective left .R-modules. Further applications are also given to provide new criteria for artinian rings with self-duality and artinian Pi-rings to be of finite representation type, which imply in particular the validity of the pure semisimple conjecture for these classes of rings.     

Indecomposable Modules and Gelfand Rings

It is proved that a commutative ring is clean if and only if it is Gelfand with a totally disconnected maximal spectrum. It is shown that each indecomposable module over a commutative ring R satisfies a finite condition if and only if R _P is an Artinian valuation ring for each maximal prime ideal P. Commutative rings for which each indecomposable module has a local endomorphism ring are studied. These rings are clean and elementary divisor rings. It is shown that each commutative ring R with a Hausdorff and totally disconnected maximal spectrum is local-global. Moreover, if R is arithmetic, then R is an elementary divisor ring.     

An Essential Extension with Nonisomorphic Ring Structures II

We construct a ring R with R = Q(R), the maximal right ring of quotients of R, and a right R-module essential extension S _R of R _R such that S has several distinct isomorphism classes of compatible ring structures. It is shown that under one class of these compatible ring structures, the ring S is not a QF-ring (in fact S is not even a right FI-extending ring), while under all other remaining classes of the ring structures, the ring S is QF. We demonstrate our results by an application to a finite ring.     

Injective Modules over Prüfer v-Multiplication Domains

Let R be an integral domain with quotient field F. It is shown that R is a strongly discrete Prüfer v-multiplication domain if and only if there exists a bijection between the set of the prime w-ideals and the set of isomorphism classes of GV-torsionfree indecomposable injective R-modules and every indecomposable injective R-module, viewed as a module over its endomorphism ring, is uniserial. It is also shown that the w-closure of any GV-torsionfree homomorphic image of F is injective if and only if R is a Prüfer v-multiplication domain satisfying an almost maximality-type property.     

On Summand Ideals in Commutative Reduced Rings

Abstract

Summand intersection property (SIP) and summand sum property (SSP) of modules are studied in Garcia [Garcia, J. L. (1989). Properties of direct summands of modules. Comm. Algebra 17(1):73–92] and Wilson [Wilson (1986). Modules with the summand intersection property. Comm. Algebra 14(1):21–38] respectively, and these properties for C(X) are studied in Azarpanah [Azarpanah, F. (1999). Sum and intersection of summand ideals in C(X). Comm. Algebra 27(11): 5549–5560]. In this paper we give some topological characterizations of these properties in the case of a commutative reduced ring R. It is shown that R has SIP if and only if every intersection of clopen subsets of Spec(R) has a closed interior. This characterization shows that R has SIP whenever Spec(R) is locally connected or extremally disconnected. It is also shown that R has SSP if and only if Spec(R) has only finitely many components. If 𝒮 is a strong subspace of Spec(R), we can replace 𝒮 with Spec(R) and all of the above results remain valid. In semiprimitive Gelfand rings, SIP and SSP are equivalent.     

Centers in domains with quadratic growth

Let F be a field, and let R be a finitely-generated F-algebra, which is a domain with quadratic growth. It is shown that either the center of R is a finitely-generated F-algebra or R satisfies a polynomial identity (is PI) or else R is algebraic over F. Let r ∈ R be not algebraic over F and let C be the centralizer of r. It is shown that either the quotient ring of C is a finitely-generated division algebra of Gelfand-Kirillov dimension 1 or R is PI.     

Central Sets and Radii of the Zero-Divisor Graphs of Commutative Rings

For a commutative ring R with identity, the zero-divisor graph, Γ(R), is the graph with vertices the nonzero zero-divisors of R and edges between distinct vertices x and y whenever xy = 0. This article gives a proof that the radius of Γ(R) is 0, 1, or 2 if R is Noetherian. The center union {0} is shown to be a union of annihilator ideals if R is Artinian. The diameter of Γ(R) can be determined once the center is identified. If R is finite, then the median is shown to be a subset of the center. A dominating set of Γ(R) is constructed using elements of the center when R is Artinian. It is shown that for a finite ring R ? ?₂ × F for some finite field F, the domination number of Γ(R) is equal to the number of distinct maximal ideals of R. Other results on the structure of Γ(R) are also presented.     

Affine Ebenen und Tupelsysteme

Zusammenfassung Es werden zwei neue Koordinatisierungsmethoden für affine Ebenen gegeben, und zwar durch homogene bzw. stark balancierte Tupelsysteme. Ihre Brauchbarkeit wird u.a. demonstriert an einer Charakterisierung der Kollineationsgruppen einer affinen Ebene mit der Eigenschaft (R) Für alle i l enthält der Stabilisator von i nur Perspektivitäten mit Zentrum i. Außerdem wird der Satz bewiesen: Für eine Elation der geraden Ordnung d in einer projektiven Ebene der Ordnung v gilt 2d|v. Dieses Resultat impliziert schon bekannte Ergebnisse von Hughes (d=2) und Hall.

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We give two new coordinatization methods for affine planes by means of homogeneous (resp. strong balanced) tuple systems. Their usefulness is demonstrated by proving a characterization of collineation groups of an affine plane with the property (R): The stabilizer of every i l contains only perspectivities with centre i. Besides, we prove: If a projective plane of order v contains an elation of even order d then 2d|v. This implies known results of Hughes (d=2) and Hall.

     

C-Pure Projective Modules

This paper investigates the structure of cyclically pure (or C-pure) projective modules. In particular, it is shown that a ring R is left Noetherian if and only if every C-pure projective left R-module is pure projective. Also, over a left hereditary Noetherian ring R, a left R-module M is C-pure projective if and only if M = N ⊕ P, where N is a direct sum of cyclic modules and P is a projective left R-module. The relationship C-purity with purity and RD-purity are also studied. It is shown that if R is a local duo-ring, then the C-pure projective left R-modules and the pure projective left R-modules coincide if and only if R is a principal ideal ring. If R is a left perfect duo-ring, then the C-pure projective left R-modules and the pure projective left R-modules coincide if and only if R is left Köthe (i.e., every left R-module is a direct sum of cyclic modules). Also, it is shown that for a ring R, if every C-pure projective left R-module is RD-projective, then R is left Noetherian, every p-injective left R-module is injective and every p-flat right R-module is flat. Finally, it is shown that if R is a left p.p-ring and every C-pure projective left R-module is RD-projective, then R is left Noetherian hereditary. The converse is also true when R is commutative, but it does not hold in the noncommutative case.     

What do multistep methods approximate?

Summary Standard analysis of multistep methods for ODE's assumes the application of an initialization routine that generates the starting points. Here ak-step method is considered directly as a mappingR ^kn R ⁿ . It is shown to approximate a mapping which is expressible directly in terms of the flow of the vector field. Some useful properties of that mapping are shown and for strictly stable methods these are applied to the question of invariant circles near a hyperbolic periodic solution.     

Polynomial Rings Over Symmetric Rings Need Not be Symmetric

Let R be a ring with identity. The polynomial ring over R is denoted by R[x] with x its indeterminate. It is shown that polynomial rings over symmetric rings need not be symmetric by an example.