scholarly journals Ideals on the Quantum Plane’s Jet Space

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 352
Author(s):  
Andrey Glubokov
Keyword(s):  
Prime Ideals ◽  
Jet Space ◽  
Quantum Plane ◽  
Prime Spectrum ◽  
Jet Spaces

The goal of this paper is to introduce some rings that play the role of the jet spaces of the quantum plane and unlike the quantum plane itself possess interesting nontrivial prime ideals. We will prove some results (Theorems 1–4) about the prime spectrum of these rings.

A localic approach to minimal prime spectra

1988 ◽  
Vol 103 (1) ◽  
pp. 47-53 ◽  
Author(s):  
Sun Shu-Hao
Keyword(s):  
Distributive Lattice ◽  
Prime Ideals ◽  
Zariski Topology ◽  
Topological Properties ◽  
Prime Spectrum ◽  
Minimal Prime

Throughout this paper, A will denote a distributive lattice with 0 and 1; we shall write spec A for the prime spectrum of A (i.e. the set of prime ideals of A, with the Stone–Zariski topology), and max A, min A for the subspaces of spec A consisting of maximal and minimal prime ideals respectively. These two subspaces have rather different topological properties: max A is always compact, but not always Hausdorff (indeed, any compact T1-space can occur as max A for some A), and min A is always Hausdorff (in fact zero-dimensional), but not always compact. (For more information on max A and min A, see Simmons[3].)

scholarly journals PRIME SPECTRA OF AMBISKEW POLYNOMIAL RINGS

2018 ◽  
Vol 61 (1) ◽  
pp. 49-68
Author(s):  
CHRISTOPHER D. FISH ◽  
DAVID A. JORDAN
Keyword(s):  
Positive Integer ◽  
Polynomial Algebra ◽  
Central Element ◽  
Universal Enveloping Algebra ◽  
Polynomial Rings ◽  
Prime Ideals ◽  
Closed Field ◽  
Prime Spectrum ◽  
Enveloping Algebra ◽  
Root Of Unity

AbstractWe determine sufficient criteria for the prime spectrum of an ambiskew polynomial algebra R over an algebraically closed field 𝕂 to be akin to those of two of the principal examples of such an algebra, namely the universal enveloping algebra U(sl2) (in characteristic 0) and its quantization Uq(sl2) (when q is not a root of unity). More precisely, we determine sufficient criteria for the prime spectrum of R to consist of 0, the ideals (z − λ)R for some central element z of R and all λ ∈ 𝕂, and, for some positive integer d and each positive integer m, d height two prime ideals P for which R/P has Goldie rank m.

CPI-Extensions: Overrings of Integral Domains with Special Prime Spectrums

1977 ◽  
Vol 29 (4) ◽  
pp. 722-737 ◽  
Author(s):  
Monte B. Boisen ◽  
Philip B. Sheldon
Keyword(s):  
Topological Space ◽  
Partially Ordered Set ◽  
Integral Domain ◽  
Ordered Set ◽  
Prime Ideals ◽  
Zariski Topology ◽  
Prime Spectrum ◽  
Integral Domains ◽  
Partially Ordered ◽  
Set Inclusion

Throughout this paper the term ring will denote a commutative ring with unity and the term integral domain will denote a ring having no nonzero divisors of zero. The set of all prime ideals of a ring R can be viewed as a topological space, called the prime spectrum of R, and abbreviated Spec (R), where the topology used is the Zariski topology [1, Definition 4, § 4.3, p. 99]. The set of all prime ideals of R can also be viewed simply as aposet - that is, a partially ordered set - with respect to set inclusion. We will use the phrase the pospec of R, or just Pospec (/v), to refer to this partially ordered set.

scholarly journals DIFFERENTIAL-ALGEBRAIC JET SPACES PRESERVE INTERNALITY TO THE CONSTANTS

2015 ◽  
Vol 80 (3) ◽  
pp. 1022-1034
Author(s):  
ZOE CHATZIDAKIS ◽  
MATTHEW HARRISON-TRAINOR ◽  
RAHIM MOOSA
Keyword(s):  
Algebraic Variety ◽  
Tangent Bundle ◽  
Jet Space ◽  
Analytic Geometry ◽  
Jet Spaces ◽  
Generic Point ◽  
Generic Type ◽  
Finite Dimensional ◽  
Generic Behaviour

AbstractSuppose p is the generic type of a differential-algebraic jet space to a finite dimensional differential-algebraic variety at a generic point. It is shown that p satisfies a certain strengthening of almost internality to the constants. This strengthening, which was originally called “being Moishezon to the constants” in [9] but is here renamed preserving internality to the constants, is a model-theoretic abstraction of the generic behaviour of jet spaces in complex-analytic geometry. An example is given showing that only a generic analogue holds in the differential-algebraic case: there is a finite dimensional differential-algebraic variety X with a subvariety Z that is internal to the constants, such that the restriction of the differential-algebraic tangent bundle of X to Z is not almost internal to the constants.

ON THE PRIME SPECTRUM OF THE ENVELOPING ALGEBRA AND CHARACTERISTIC VARIETIES

2007 ◽  
Vol 06 (03) ◽  
pp. 369-383 ◽  
Author(s):  
NIKOLAOS PAPALEXIOU
Keyword(s):  
Topological Space ◽  
Lie Algebra ◽  
Prime Ideals ◽  
Prime Spectrum ◽  
Semisimple Lie Algebra ◽  
Characteristic Variety ◽  
Enveloping Algebra ◽  
Commutative Algebras ◽  
Order Relations ◽  
Characteristic Varieties

Let 𝔤 be a semisimple Lie algebra and U(𝔤), its enveloping algebra. A problem in the theory of non-commutative algebras is the description of the set Spec U(𝔤) of prime ideals in U(𝔤) as topological space. Using the notion of the characteristic variety as introduced by Joseph, we compute some order relations between prime ideals.

The Minimal Prime Spectrum of a Commutative Ring

1971 ◽  
Vol 23 (5) ◽  
pp. 749-758 ◽  
Author(s):  
M. Hochster
Keyword(s):  
Topological Space ◽  
Commutative Ring ◽  
Prime Ideals ◽  
Zariski Topology ◽  
Prime Spectrum ◽  
Topological Characterization ◽  
Completely Regular ◽  
Minimal Prime

We call a topological space X minspectral if it is homeomorphic to the space of minimal prime ideals of a commutative ring A in the usual (hull-kernel or Zariski) topology (see [2, p. 111]). Note that if A has an identity, is a subspace of Spec A (as defined in [1, p. 124]). It is well known that a minspectral space is Hausdorff and has a clopen basis (and hence is completely regular). We give here a topological characterization of the minspectral spaces, and we show that all minspectral spaces can actually be obtained from rings with identity and that open (but not closed) subspaces of minspectral spaces are minspectral (Theorem 1, Proposition 5).

scholarly journals The prime spectrum and primitive ideal space of a graph C*-algebra

2014 ◽  
Vol 25 (07) ◽  
pp. 1450070
Author(s):  
Gene Abrams ◽  
Mark Tomforde
Keyword(s):  
Disjoint Union ◽  
Satisfying Condition ◽  
Primitive Ideal ◽  
Prime Ideals ◽  
Ideal Space ◽  
Prime Spectrum ◽  
C Algebra ◽  
Primitive Ideals

We describe primitive and prime ideals in the C*-algebra C*(E) of a graph E satisfying Condition (K), together with the topologies on each of these spaces. In particular, we find that primitive ideals correspond to the set of maximal tails disjoint union the set of finite-return vertices, and that prime ideals correspond to the set of clusters of maximal tails disjoint union the set of finite-return vertices.

Numerical Solution of Constrained Systems Using Jet Spaces

2007 ◽  
Author(s):  
Jukka Tuomela ◽  
Teijo Arponen ◽  
Villesamuli Normi
Keyword(s):  
Multibody Systems ◽  
Equations Of Motion ◽  
Constrained Systems ◽  
Lagrangian Formalism ◽  
Jet Space ◽  
Jet Spaces ◽  
Conservation Of Energy ◽  
Holonomic Constraints ◽  
Motion Constraints ◽  
Spurious Oscillations

The major difficulty in simulations of constrained systems is how to avoid drift off and spurious oscillations. We describe results obtained by our method which addresses these issues. Our computational model considers differential equations in jet spaces. In case of multibody systems we use Lagrangian formalism to derive the equations of motion. Constraints and possible invariants (like conservation of energy) are taken into account by restricting the dynamics to an appropriate submanifold of a jet space. We will consider only holonomic constraints in this paper. However, nonholonomic problems can also be formulated and solved with our method.

scholarly journals PRIMES OF HEIGHT ONE AND A CLASS OF NOETHERIAN FINITELY PRESENTED ALGEBRAS

2007 ◽  
Vol 17 (07) ◽  
pp. 1465-1491 ◽  
Author(s):  
ISABEL GOFFA ◽  
ERIC JESPERS ◽  
JAN OKNIŃSKI
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Prime Ideals ◽  
Semigroup Algebras ◽  
Prime Spectrum ◽  
Maximal Orders ◽  
Identity Based ◽  
A Finite Group ◽  
The Relationship ◽  
Finitely Presented

Constructions are given of Noetherian maximal orders that are finitely presented algebras over a field K, defined by monomial relations. In order to do this, it is shown that the underlying homogeneous information determines the algebraic structure of the algebra. So, it is natural to consider such algebras as semigroup algebras K[S] and to investigate the structure of the monoid S. The relationship between the prime ideals of the algebra and those of the monoid S is one of the main tools. Results analogous to fundamental facts known for the prime spectrum of algebras graded by a finite group are obtained. This is then applied to characterize a large class of prime Noetherian maximal orders that satisfy a polynomial identity, based on a special class of submonoids of polycyclic-by-finite groups. The main results are illustrated with new constructions of concrete classes of finitely presented algebras of this type.

scholarly journals Zariski Topology

2018 ◽  
Vol 26 (4) ◽  
pp. 277-283
Author(s):  
Yasushige Watase
Keyword(s):  
Commutative Ring ◽  
Local Ring ◽  
Jacobson Radical ◽  
Commutative Rings ◽  
Ring Homomorphism ◽  
Ring Theory ◽  
Prime Ideals ◽  
Topological Spaces ◽  
Zariski Topology ◽  
Prime Spectrum

Summary We formalize in the Mizar system [3], [4] basic definitions of commutative ring theory such as prime spectrum, nilradical, Jacobson radical, local ring, and semi-local ring [5], [6], then formalize proofs of some related theorems along with the first chapter of [1]. The article introduces the so-called Zariski topology. The set of all prime ideals of a commutative ring A is called the prime spectrum of A denoted by Spectrum A. A new functor Spec generates Zariski topology to make Spectrum A a topological space. A different role is given to Spec as a map from a ring morphism of commutative rings to that of topological spaces by the following manner: for a ring homomorphism h : A → B, we defined (Spec h) : Spec B → Spec A by (Spec h)(𝔭) = h−1(𝔭) where 𝔭 2 Spec B.

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