Torus-invariant prime ideals in quantum matrices, totally nonnegative cells and symplectic leaves

The algebra of quantum matrices of a given size supports a rational torus action by automorphisms. It follows from work of Letzter and the first named author that to understand the prime and primitive spectra of this algebra, the first step is to understand the prime ideals that are invariant under the torus action. In this paper, we prove that a family of quantum minors is the set of all quantum minors that belong to a given torus-invariant prime ideal of a quantum matrix algebra if and only if the corresponding family of minors defines a non-empty totally nonnegative cell in the space of totally nonnegative real matrices of the appropriate size. As a corollary, we obtain explicit generating sets of quantum minors for the torus-invariant prime ideals of quantum matrices in the case where the quantisation parameter q is transcendental over ${\mathbb{Q}}$ .     

The lifting of determinantal prime ideals

The main results of this paper show that a perfect prime ideal generated by the maximal minors of a matrix has the equality between symbolic and ordinary powers if the ideals generated by the low order minors of the matrix have grade large enough and that any determinantal prime ideal of maximal minors with maximal grade of a matrix of homogenous forms whose 2-minors are homogeneous can be lifted to a prime determinantal ideal having the above equality. The author is partially supported by the National Basic Research Program     

Decompositions of Ideals of Minors Meeting a Submatrix

We compute the primary decomposition of certain ideals generated by subsets of minors in a generic matrix or in a generic symmetric matrix, or subsets of Pfaffians in a generic skew-symmetric matrix. Specifically, the ideals we consider are generated by minors that have at least some given number of rows and columns in certain submatrices.     

On Ideals of Minors of Matrices with Indeterminate Entries

This article studies ideals of minors of matrices whose entries are among the variables of a polynomial ring. The main result is a theorem which gives sufficient conditions for these to be prime.     

Polynomial relations among principal minors of a -matrix

The image of the principal minor map for n×n-matrices is shown to be closed. In the 19th century, Nanson and Muir studied the implicitization problem of finding all relations among principal minors when n=4. We complete their partial results by constructing explicit polynomials of degree 12 that scheme-theoretically define this affine variety and also its projective closure in . The latter is the main component in the singular locus of the 2×2×2×2-hyperdeterminant.     

On the Symmetric and Rees algebras of an ideal

Some necessary and sufficient conditions are given for the Rees and Symmetric Algebra of an ideal being canonically isomorphic. Some applications are obtained to the study of the relations between the generators of ideals which are maximal minors of a generic t by (t+1) matrix, prime ideals of finite projective dimension, almost complete intersections or d-sequences.This paper was supported by C. N. R. (Consiglio Nazionale delle Ricerche)     

Dimension and enumeration of primitive ideals in quantum algebras

In this paper, we study the primitive ideals of quantum algebras supporting a rational torus action. We first prove a quantum analogue of a Theorem of Dixmier; namely, we show that the Gelfand-Kirillov dimension of primitive factors of various quantum algebras is always even. Next we give a combinatorial criterion for a prime ideal that is invariant under the torus action to be primitive. We use this criterion to obtain a formula for the number of primitive ideals in the algebra of 2×n quantum matrices that are invariant under the action of the torus. Roughly speaking, this can be thought of as giving an enumeration of the points that are invariant under the induced action of the torus in the “variety of 2×n quantum matrices”. The first author thanks NSERC for its generous support. This research was supported by a Marie Curie Intra-European Fellowship within the 6th European Community Framework Programme held at the University of Edinburgh, by a Marie Curie European Reintegration Grant within the 7th European Community Framework Programme and by Leverhulme Research Interchange Grant F/00158/X.     

On strongly prime rings and ideals

Strongly prime rings may be defined as prime rings with simple central closure. This paper is concerned with further investigation of such rings. Various characterizations, particularly in terms of symmetric zero divisors, are given. We prove that the central closure of a strongly (semi-)prime ring may be obtained by a certain symmetric perfect one sided localization. Complements of strongly prime ideals are described in terms of strongly multiplicative sets of rings. Moreover, some relations between a ring and its multiplication ring are examined.     

3-standardness of the maximal ideal

We study a notion called n-standardness (defined by M.E. Rossi (2000) in [10] and extended in this paper) of ideals primary to the maximal ideal in a Cohen-Macaulay local ring and some of its consequences. We further study conditions under which the maximal ideal is 3-standard, first proving results for when the residue field has prime characteristic and then using the method of reduction to prime characteristic to extend the results to the equicharacteristic 0 case. As an application, we extend a result due to T. Puthenpurakal (2005) [9] and show that a certain length associated with a minimal reduction of the maximal ideal does not depend on the minimal reduction chosen.     

Boolean and distributive ordered sets: Characterization and representation by sets

Boolean ordered sets generalize Boolean lattices, and distributive ordered sets generalize distributive lattices. Ideals, prime ideals, and maximal ideals in ordered sets are defined, and some well-known theorems on Boolean lattices, such as the Glivenko-Stone theorem and the Stone representation theorem, are generalized to Boolean ordered sets. A prime ideal theorem for distributive ordered sets is formulated, and the Birkhoff representation theorem is generalized to distributive ordered sets. Fundamental are the embedding theorems for Boolean ordered sets and for distributive ordered sets.Financial support of the Grant Agency of the Czech Republic under the grant No. 201/93/0950 is gratefully acknowledged.     

PROPOSITIONAL LOGIC,FRAMES, AND FUZZY ALGEBRA

Abstract

This paper offers a new look at such things as the fuzzy subalgebras and congruences of an algebra, the fuzzy ideals of a ring or a lattice, and similar entities, by exhibiting them as the models, in the chosen frame T of truth values, of naturally corresponding propositional theories. This provides a systematic approach to the study of the partially ordered sets formed by these various entities, and we demonstrate its usefulness by employing it to derive a number of results, some old and some new, concerning these partially ordered sets. In particular, we prove they are complete lattices, algebraic or continuous, depending on whether T is algebraic or continuous, respectively (Proposition 3); they satisfy the same lattice identities for arbitrary T that hold in the case T = 2 (Corollary of Proposition 4); and they are coherent frames for any coherent T whenever this is the case for T = 2 (Proposition 6). In addition we show, generalizing a result by Makamba and Murali [10], that the familiar classical situations where the congruences of an algebra correspond to certain other entities, such as the normal subgroups of a group or the ideals of a ring, extend to the fuzzy case by proving that the corresponding propositional theories are equivalent (Proposition 2). Further, we obtain the result of Gupta and Kantroo [5] that the fuzzy radical ideals of a commutative ring with unit are the meets of fuzzy prime ideals for arbitrary continuous T in place of the unit interval, using basic facts concerning continuous frames (Proposition 7).     

Multiplicative sets, left strongly prime ideals, and left strongly prime radicals in rings

Left commutative multiplicative sets {ie397-01} for an associative ring R are defined. In particular, this notion includes commutative multiplicative sets of the associative ring. We also define the notion of a left {ie397-02}-ideal and prove that each left {ie397-03}-ideal, maximal with respect to being disjoint from {ie397-04}, is left strongly prime. Using a technique developed for insulators for a left ideal, we also characterize the left strongly prime radical of a left ideal of the ring R, which was known only for two-sided ideals.     

Commutation relations among quantum minors in

In this paper we compute explicit formulas for the commutation relations between any two quantum minors in the quantum matrix bialgebra O_q(M_n(k)). The product of any two minors is expressed as linear combination of products of minors strictly less in certain orderings.     

The Extensional Structure of Commutative Noetherian Rings

Abstract

We show that finitely generated modules over a commutative Noetherian ring can be classified, up to isomorphism of submodule series, in a manner analogous to the classification of integers as products of prime numbers. In outline, two such modules have isomorphic submodule series if and only if 1) the set of minimal associated prime ideals of these modules coincide, 2) the multiplicities of these modules at these prime ideals coincide, and 3) the modules represent the same element in a certain group corresponding to the above set of prime ideals. Regarding condition 3), we show that, in the very special case that the ring is a Dedekind domain, the group corresponding to the prime ideal (0) is the ideal class group of the ring.     

IDEALS OF MINORS DEFINING GENERIC SINGULARITIES AND THEIR GRöBNER BASES

In this article, using the local parametric equations of a generic projection π of a smooth projective variety X, at an analytically irreducible singular point y of X′ = π(X), the defining ideals J and J′ of X′ and its singular locus at y are expressed as ideals of maximal and sub-maximal minors of certain Sylvester matrix @. The proof is obtained by a convenient reduction of @ to a “generic pluri-circulant matrix” P and the construction of minimal Gröbner bases for the ideal of t-minors of P and for the ideals J and J′. The depth of local rings of X′ and Sing (X′) at y are also computed in terms of the multiplicity at y.     

On right S-Noetherian rings and S-Noetherian modules

In this paper we study right S-Noetherian rings and modules, extending notions introduced by Anderson and Dumitrescu in commutative algebra to noncommutative rings. Two characterizations of right S-Noetherian rings are given in terms of completely prime right ideals and point annihilator sets. We also prove an existence result for completely prime point annihilators of certain S-Noetherian modules with the following consequence in commutative algebra: If a module M over a commutative ring is S-Noetherian with respect to a multiplicative set S that contains no zero-divisors for M, then M has an associated prime.     

Linked Primes and Orders in Artinian Rings

It is shown that prime ideals of a Noetherian ring are linked if and only if certain corresponding prime ideals are linked in an associated Artinian ring. Furthermore, it is shown that there is a canonical linking ideal, which can be found by using a construction based on middle annihilator ideals.     

The quantum euclidean algebra and its prime spectrum

For the quantum Euclidean algebra, its prime, completely prime and maximal spectra are described (together with inclusions of prime ideals). The centre is generated by two algebraically independent elements (one is quadratic and the other is cubic).