Birational rowmotion on a rectangle over a noncommutative ring

Let B be an associative ring with identity, A a subring of B containing the identity of B. If B is commutative then it is customary to define an element b of B to be integral over A if it satisties an equation of the form for some a1, a2, …, an A. This definition does not generalize readily to the case when B is non-commutative. Van der Waerden ([11], p. 75) defines b ∈ B to be integral over A if all powers of b belong to a finite A-module. This definition is quite satisfactory when A satisfies the ascending chain condition for left ideals, but in the general case this type of integrity is not necessarily transitive, even when B is commutative. Krull [6] calls an element b ∈ B which satisfies the above condition almost integral over A (but he only considers the commutative case). The subset Ā of B consisting of all almost integral elements over A is called the complete integral closure of A in B. If Ā = A, A is said to be completely integrally closed in B. More recently (in [3]), Gilmer and Heinzer (see also Bourbaki, [1]) have discussed these properties in the commutative case and have shown that the complete integral closure of A in B need not be completely integrally closed in B. If B is not commutative, the set A of elements of B almost integral over A, may not even form a ring. In [5] p. 122, Jacobson uses a definition equivalent to Van der Waerden's for the non-commutative case but the definition applies only for a very restricted class of rings.

Since the late sixties, various Auslander regularity conditions have been widely investigated in both commutative and noncommutative cases, e.g., [Ba2], [FGR], [Roos], [Lev], etc .... In particular, the dimension and multiplicity theory for modules, the pure (holonomic) module theory over noncommutative Auslander regular rings have been developed in the study of rings of differential operators and enveloping algebras, e.g., [Ka], [Be], [B-B], [K-T], etc ... A typical result concerning purity is the so-called Gabber-Kashiwara theorem (see [Gi] Theorem V8) which has been used to: (1). determine the existence of holonomic submodules in D-modules and the existence of a b-function; (2). determine the equidimensionality of the characteristic variety of a simple module over the enveloping algebra of a Lie algebra. In this chapter we first present a general study of Auslander regularity of filtered rings and graded rings; Then we generalize the important Gabber-Kashiwara theorem and Roos theorem to Zariski rings with commutative associated graded rings including most of the examples given in Ch.II. §2; and consequently the pure module theory over Zariski ring is developed as well.KeywordsExact SequenceSpectral SequenceNoetherian RingRegular RingProjective ResolutionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In this paper, we consider representations of Coxeter groups over a path algebra, R, introduced by Dyer in the study of root systems over non-commutative rings. We answer a question posed by Dyer about the multiplicative properties of R, showing that it is “almost a domain.” We also show that R cam be embedded in a matrix ring over a free product of extension fields of the rational numbers and rings of Laurent polynomials.

In 1968–1969, A. V. Jategaonkar published his famous constructions of left but not right noetherian rings that provided counterexamples to several important conjectures of that era. These examples, and others like them, seemed to indicate that, in general, the task of completely understanding the structure of indecomposable injective modules over one-sided noetherian rings was hopeless. In this paper I show how to deduce by natural methods, directly from the known description of these rings and their properties, explicit computational descriptions of the indecomposable injective left modules over Jategaonkar's rings. I use these explicit descriptions to answer some simple structural questions about the indecomposables. The structure of indecomposable injective modules over a commutative noetherian ring has been well understood since the work of Matlis[10] in the 1950's. There is a natural correspondence between the indecomposable injectives and the localizations of the ring, and the structure of ...

The spin-one Duffin-Kemmer-Petiau oscillator in uniform magnetic field is studied in noncommutative formalism. The corresponding energy is obtained and thereby the corresponding thermal properties are obtained for both commutative and noncommutative cases.

We develop the superfield approach to the effective potential in three dimensions and calculate the one-loop and two-loop Kählerian effective potential in commutative and noncommutative cases.

This chapter is an overview of the arguments relating symmetry, experimental data, and inference that appear in the theory, methods, and applications of dihedral analysis. The example introduced here is based on the only commutative case (n = 2) and yet it includes all the basic elements present in the non-commutative (n > 2) cases.

Multispectral Hg 1− xCd xTe photovoltaic detectors∗

We give a complete characterization of bipartite graphs having tree-like Galois lattices. We prove that the poset obtained by deleting bottom and top elements from the Galois lattice of a bipartite graph is tree-like if and only if the graph is a Bipartite Distance Hereditary graph. We show that the lattice can be realized as the containment relation among directed paths in an arborescence. Moreover, a compact encoding of Bipartite Distance Hereditary graphs is proposed, that allows optimal time computation of neighborhood intersections and maximal bicliques.

Roof truss systems and wall panels are two examples of popular structural components made by cold-formed steel (CFS) which is classified as a steel-based building material available in a wide range of shapes, and thicknesses. The use of CFS is increasing due to its many desirable properties, including a high strength-to-weight ratio, low weight, resistance to corrosion, and fast installation. In contrast, the thin-surfaced and opened section of CFS buckles easily when utilised as a beam or column. When the load is applied directly, the CFS section fails in modes such as web crippling, torsion, and buckling. As a result, the new I-section with top and bottom flange elements was produced by using CFS channel section, and timber board from the type of oriented strand board (OSB) as a web element. The study aimed to determine the initial structural characteristics of the built-up I-section CFS with OSB short column. A single or double web element made by yellow and dark brown OSB was used to construct the I-section for determining the initial structural characteristics. The I-section with a double web of yellow OSB showed the highest value of ultimate load and compressive strength as compared to other specimens in the range of 8.00% to 14.00%.

Cold-formed steel roof truss systems that use complex hat shape members for both top and bottom chord elements are a growing trend in the North America steel framing industry. When designing cold-formed steel sections, a structural engineer typically tries to improve the local buckling behavior of the cold-formed steel elements. The complex hat shape has proven to limit the negative influence of local buckling. However, a distortional buckling mode can be the control mode of failure in the design for the chord member with an intermediate unbraced length. The chord member may be subjected to both bending and compressive load because of the continuity of the top and bottom chord members. These members are not typically braced between each panel point in a truss. Numerical analyses using finite strip and finite element procedures were developed to compare with experimental results. A parametric study on geometric imperfection was also conducted to investigate the factors that affect the ultimate strength behavior of a particular complex hat shape. Better understanding of the flexural behavior of these complex hat shapes is necessary to obtain efficient, safe designs of a truss system. The results of these analyses will be presented in the paper.

On the invertibility of ideals in orders

ABSTRACTWe extend the definition of a piecewise Noetherian ring to the noncommutative case, and investigate various properties of such rings. In particular, we show that a ring with Krull dimension is piecewise Noetherian. Certain fully bounded piecewise Noetherian rings have Gabriel dimension and exhibit the Gabriel correspondence between prime ideals and indecomposable injective modules.

Article Free Access Share on Voronoi diagrams and containment of families of convex sets on the plane Authors: M. Abellanas Universidad Politécnica de Madrid, Spain Universidad Politécnica de Madrid, SpainView Profile , G. Hernandez Universidad Politécnica de Madrid, Spain Universidad Politécnica de Madrid, SpainView Profile , R. Klein Fern Universität Hagen, Germany Fern Universität Hagen, GermanyView Profile , V. Neumann-Lara Universidad Nacional Autonoma de México, Mexico Universidad Nacional Autonoma de México, MexicoView Profile , J. Urrutia University of Ottawa, Canada University of Ottawa, CanadaView Profile Authors Info & Claims SCG '95: Proceedings of the eleventh annual symposium on Computational geometrySeptember 1995 Pages 71–78https://doi.org/10.1145/220279.220287Published:01 September 1995Publication History 3citation259DownloadsMetricsTotal Citations3Total Downloads259Last 12 Months12Last 6 weeks4 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF

1. This paper contains a generalization in the commutative case of the structure theorem of W. Ambrose for H*-algebras. An immediate result of Ambrose's axioms is that the orthogonal complement of an ideal is an ideal of the same nature. This fact suggests thinking in terms of a Banach algebra whose conjugate space is a Banach algebra such that the annihilator of an ideal in one algebra is an ideal in the other algebra. This is the essential idea in the definition of a commutative, semi-simple, reflexive Banach algebra given in ?2. This section shows that such an algebra is essentially an 1,-space in which a product has been defined by means of co-ordinate multiplication. ?3 contains a few remarks and examples concerning the author's conjecture that a similar structure theorem must exist in the noncommutative case. The term annihilator will be constantly needed in both its ring and vector space sense. The phrase ring annihilator will be used to distinguish the use of the term in the ring sense from its use in the vector space sense. B' will be used to denote the conjugate space of the given algebra B. Greek letters e, 8, and p will be used to denote complex numbers. Elements in B will be denoted by letters x, y, and z, while elements in B' will be denoted by f, g, and h. The value of the linear functionalf at the point x will be denoted by (x, f). Projections in B will be denoted by means of the letter E, while those in B' will be denoted by means of E'. The following theorems from the general theory will be used.