We show how the usual algorithms valid over Euclidean domains, such as the Hermite Normal Form, the modular Hermite Normal Form and the Smith Normal Form can be extended to Dedekind rings. In a sequel to this paper, we will explain the use of these algorithms for computing in relative extensions of number fields. The goal of this paper is to explain how to generalize to a Dedekind domain R many...

This paper contains the results collected so far on polynomial composites in terms of many basic algebraic properties. Since it is a structure, for monoid domains come here and there. The second part relationship between theory composites, Galois nilpotents. third this shows us some crypto systems. We find generalizations known ciphers taking into account infinite alphabet using simple methods....

If every subring of an integral domain is atomic, we say that the latter hereditarily atomic. In this paper, study atomic domains. First, characterize when certain direct limits Dedekind domains are in terms overrings. Then use characterization to determine fields On other hand, investigate hereditary atomicity context rings polynomials and Laurent polynomials, characterizing whose respectively...

In the study of hereditary Noetherian rings, it is clear that hereditary Noetherian prime rings will play a central role (see, for example, [12]). Here we study the (two-sided) ideals of an hereditary Xoetherian prime ring and, as a consequence, ascertain the structure of factor rings and torsion modules. The torsion theory represents a generalization of similar results about Dedekind prime rin...

Remark 1.6. In [Mor06], this result is stated without proof. Our goal here is to indicate that the result follows without any of the machinations appearing in §3.2 of [Mor06] (where the result is stated). Ayoub has shown that if the Krull dimension of k is ≥ 2, then the unstable A-connectivity property will fail, thus the strong A-invariance property cannot hold over 2-dimensional rings. Howeve...

This paper reports on a construction of the real numbers in the ProofPower implementation of the HOL logic. Since the original construction was implemented, some major improvements to the ideas have been discovered. The improvements involve some entertaining mathematics: it turns out that the Dedekind cuts provide many routes one can travel to get from the ring of integers, Z, to the field of r...

In this paper, I present a new decision procedure for the ideal membership problem for polynomial rings over principal domains using discrete valuation domains. As a particular case, I solve a fundamental algorithmic question in the theory of multivariate polynomials over the integers called “Kronecker’s problem”, that is the problem of finding a decision procedure for the ideal membership prob...

We prove a generalization of the Flat Cover Conjecture by showing for any ring R that (1) each (right R-) module has a Ker Ext(−, C)-cover, for any class of pure-injective modules C, and that (2) each module has a Ker Tor(−,B)-cover, for any class of left R-modules B. For Dedekind domains, we describe Ker Ext(−, C) explicitly for any class of cotorsion modules C; in particular, we prove that (1...

In these notes we will first define projective modules and prove some standard properties of those modules. Then we will classify finitely generated projective modules over Dedekind domains Remark 0.1. All rings will be commutative with 1. 1. Projective modules Definition 1.1. Let R be a ring and let M be an R-module. Then M is called projective if for all surjections p : N → N ′ and a map f : ...

The first step in investigating colour symmetries for periodic and aperiodic systems is the determination of all colouring schemes that are compatible with the symmetry group of the underlying structure, or with a subgroup of it. For an important class of colourings of planar structures, this mainly combinatorial question can be addressed with methods of algebraic number theory. We present the ...