In this paper, we will investigate a question of Anderson–Zafrullah of whether an LPI domain (every nonzero locally principal ideal is invertible) has finite prime character and characterize pullbacks that are LPI domains. 相似文献
We give necessary and sufficient conditions that the pullback of a conductor square be a chain ring (i.e., a ring whose ideals are totally ordered by inclusion). We also give necessary and sufficient conditions that the pullback of a conductor square be an arithmetical ring (i.e., a ring which is locally a chain ring at every maximal ideal). For any integral domain D with field of fractions K, we characterize all Prüfer domains R between D[X] and K[X] such that the conductor C of K[X] into R is nonzero. As an application, we show that for n ≥ 2, such a ring R has the n-generator property (every finitely generated ideal can be generated by n elements) if and only if R/C has the same property. 相似文献
Let be an integral domain and let be a nonzero polynomial in . The content of is the ideal generated by the coefficients of . The polynomial is called Gaussian if for all . It is well known that if is an invertible ideal, then is Gaussian. In this note we prove the converse.
We prove a new extension result for QB-rings that allows us to examine extensions of rings where the ideal is purely infinite and simple. We then use this result to explore various constructions that provide new examples of QB-rings. More concretely, we show that a surjective pullback of two QB-rings is usually again a QB-ring. Specializing to the case of an extension of a semi-prime ideal I of a unital ring R, the pullback setting leads naturally to the study of rings whose multiplier rings are QB-rings. For a wide class of regular rings, we give necessary and sufficient conditions for their multiplier rings to be QB-rings. Our analysis is based on the study of extensions and the use of nonstable K-theoretical techniques.
Presented by S. MontgomeryMathematics Subject Classification (2000) 16D30.Gert K. Pedersen: Deceased 15 March 2004.Partially supported by the DGI and European Regional Development Fund, jointly, through Project BFM 2002-01390, the Comissionat per Universitats i Recerca de la Generalitat de Catalunya and the Danish Research Council. 相似文献
Let A be a commutative ring and E a nonzero A-module. Necessary and sufficient conditions are given for the trivial ring extension R of A by E to be either arithmetical or Gaussian. The possibility for R to be Bézout is also studied, but a response is only given in the case where pSpec(A) (a quotient space of Spec(A)) is totally disconnected. Trivial ring extensions which are fqp-rings are characterized only in the local case. To get a general result we intoduce the class of fqf-rings satisfying a weaker property than fqp-ring. Moreover, it is proven that the finitistic weak dimension of a fqf-ring is 0, 1 or 2 and its global weak dimension is 0, 1, or ∞. 相似文献
The aim of this paper is to take up again the study done in previous papers, to the case where the integrand possesses an
algebraic singularity within the interval of integration. The singularities or poles close to the interval of integration
considered in this paper are only real or purely imaginary.
This revised version was published online in August 2006 with corrections to the Cover Date. 相似文献
Let D be an integral domain with quotient field K. A Bhargava ring over D is defined to be 𝔹x(D): = {f ∈ K[X] | ? a ∈ D, f(xX + a) ∈ D[X]}, where x ∈ D. A Bhargava ring over D is a subring of the ring of integer-valued polynomials over D. In this article, we study the prime ideal structure and calculate the Krull and valuative dimension of Bhargava rings over a general domain D. 相似文献
It is well known that Gaussian polynomials (i.e., q-binomials) describe the distribution of the statistic on monotone paths in a rectangular grid. We introduce two new statistics, and ; attach “ornaments” to the grid that scramble the values of in specific fashion; and re-evaluate these statistics, in order to argue that all scrambled versions of the statistic are equidistributed with . Our main result is a representation of the generating function for the bi-statistic as a new, two-variable Vandermonde convolution of the original Gaussian polynomial. The proof relies on explicit bijections between differently ornated paths. 相似文献
In classical theorems on the convergence of Gaussian quadrature formulas for power orthogonal polynomials with respect to a weight w on I =(a,b),a function G ∈ S(w):= { f:∫I | f(x)| w(x)d x < ∞} satisfying the conditions G 2j(x) ≥ 0,x ∈(a,b),j = 0,1,...,and growing as fast as possible as x → a + and x → b,plays an important role.But to find such a function G is often difficult and complicated.This implies that to prove convergence of Gaussian quadrature formulas,it is enough to find a function G ∈ S(w) with G ≥ 0 satisfying sup n ∑λ0knG(xkn) k=1 n<∞ instead,where the xkn ’s are the zeros of the n th power orthogonal polynomial with respect to the weight w and λ0kn ’s are the corresponding Cotes numbers.Furthermore,some results of the convergence for Gaussian quadrature formulas involving the above condition are given. 相似文献
It is well known that Gaussian cubature rules are related to multivariate orthogonal polynomials. The cubature rules found in the literature use common zeroes of some linearly independent set of products of basically univariate polynomials. We show how a new family of multivariate orthogonal polynomials, so-called spherical orthogonal polynomials, leads to symbolic-numeric Gaussian cubature rules in a very natural way. They can be used for the integration of multivariate functions that in addition may depend on a vector of parameters and they are exact for multivariate parameterized polynomials. Purely numeric Gaussian cubature rules for the exact integration of multivariate polynomials can also be obtained.We illustrate their use for the symbolic-numeric solution of the partial differential equations satisfied by the Appell function F2, which arises frequently in various physical and chemical applications. The advantage of a symbolic-numeric formula over a purely numeric one is that one obtains a continuous extension, in terms of the parameters, of the numeric solution. The number of symbolic-numeric nodes in our Gaussian cubature rules is minimal, namely m for the exact integration of a polynomial of homogeneous degree 2m−1.In Section 1 we describe how the symbolic-numeric rules are constructed, in any dimension and for any order. In Sections 2, 3 and 4 we explicit them on different domains and for different weight functions. An illustration of the new formulas is given in Section 5 and we show in Section 6 how numeric cubature rules can be derived for the exact integration of multivariate polynomials. From Section 7 it is clear that there is a connection between our symbolic-numeric cubature rules and numeric cubature formulae with a minimal (or small) number of nodes. 相似文献
More general and stronger estimations of bounds for the fundamental functions of Hermite interpolation of high order on an arbitrary system of nodes on infinite intervals are given. Based on this result, convergence of Gaussian quadrature formulas for Riemann–Stieltjes integrable functions on an arbitrary system of nodes on infinite intervals is discussed. 相似文献
We analyze polynomials Pn that are biorthogonal to exponentials
, in the sense that
Here α>−1. We show that the zero distribution of Pn as n→∞ is closely related to that of the associated exponent polynomial
More precisely, we show that the zero counting measures of {Pn(−4nx)}n=1∞ converge weakly if and only if the zero counting measures of {Qn}n=1∞ converge weakly. A key step is relating the zero distribution of such a polynomial to that of the composite polynomial
Let R(X) = Q[x1, x2, ..., xn] be the ring of polynomials in the variables X = {x1, x2, ..., xn} and R*(X) denote the quotient of R(X) by the ideal generated by the elementary symmetric functions. Given a Sn, we let g
In the late 1970s I. Gessel conjectured that these monomials, called the descent monomials, are a basis for R*(X). Actually, this result was known to Steinberg [10]. A. Garsia showed how it could be derived from the theory of Stanley-Reisner Rings [3]. Now let R(X, Y) denote the ring of polynomials in the variables X = {x1, x2, ..., xn} and Y = {y1, y2, ..., yn}. The diagonal action of Sn on polynomial P(X, Y) is defined as
Let R(X, Y) be the subring of R(X, Y) which is invariant under the diagonal action. Let R*(X, Y) denote the quotient of R(X, Y) by the ideal generated by the elementary symmetric functions in X and the elementary symmetric functions in Y. Recently, A. Garsia in [4] and V. Reiner in [8] showed that a collection of polynomials closely related to the descent monomials are a basis for R*(X, Y). In this paper, the author gives elementary proofs of both theorems by constructing algorithms that show how to expand elements of R*(X) and R*(X, Y) in terms of their respective bases. 相似文献
B-construction is a way of obtaining a graded algebra from the triple consisting of an additive category, an object, and an autoequivalence, while C-construction is a way of obtaining an algebra (without unity) from the pair consisting of an additive category and a set of objects. In this article, we study and compare three important classes of algebras in noncommutative algebraic geometry and representation theory of finite dimensional algebras, namely, quantum polynomial algebras, preprojetive algebras and trivial extensions, via these constructions. 相似文献