Strongly Hilbert Modules

We will provide some results on Hilbert modules, namely an equivalent condition forfaithful Noetherian modules to be Hilbert. Then, we will generalize the notion of a Hilbertrings and modules to create the concept of C-Hilbert rings and modules. Finally, to providemore examples of C-Hilbert modules, we will take the notion of strongly Hilbert rings andextend them to strongly Hilbert modules

Computing representations for radicals of finitely generated differential ideals

This paper deals with systems of polynomial differential equations, ordinary or with partial derivatives. The embedding theory is the differential algebra of Ritt and Kolchin. We describe an algorithm, named Rosenfeld-Gröbner, which computes a representation for the radical P of the differential ideal generated by any such system S. The computed representation constitutes a normal simplifier for the equivalence relation modulo P (it permits to test membership in P). It permits also to compute Taylor expansions of solutions of S. The algorithm is implemented within a package in MAPLE V

Computing a Gröbner basis of a polynomial ideal over a Euclidean domain

AbstractAn algorithm for computing a Gröbner basis of a polynomial ideal over a Euclidean domain is presented. The algorithm takes an ideal specified by a finite set of polynomials as its input; it produces another finite basis of the same ideal with the properties that using this basis, every polynomial in the ideal reduces to 0 and every polynomial in the polynomial ring reduces to a unique normal form. The algorithm is an extension of Buchberger's algorithms for computing Gröbner bases of polynomial ideals over an arbitrary field and over the integers as well as our algorithms for computing Gröbner bases of polynomial ideals over the integers and the Gaussian integers. The algorithm is simpler than other algorithms for polynomial ideals over a Euclidean domain reported in the literature; it is based on a natural way of simplifying polynomials by another polynomial using Euclid's division algorithm on the coefficients in polynomials. The algorithm is illustrated by showing how to compute Gröbner bases for polynomial ideals over the integers, the Gaussian integers as well as over algebraic integers in quadratic number fields admitting a division algorithm. A general theorem exhibiting the uniqueness of a reduced Gröbner basis of an ideal, determined by an admissible ordering on terms (power products) and other conditions, is discussed