Factorization Of Ideals Research Articles

N-absorbing ideal factorization in commutative rings

In this article, we show that Mori domains, pseudo-valuation domains, and n-absorbing ideals, the three seemingly unrelated notions in commutative ring theory, are interconnected. In particular, we prove that an integral domain R is a Mori locally pseudo-valuation domain if and only if each proper ideal of R is a finite product of 2-absorbing ideals of R. Moreover, every ideal of a Mori locally almost pseudo-valuation domain can be written as a finite product of 3-absorbing ideals. To provide concrete examples of such rings, we study rings of the form A + XB [ X ] where A is a subring of a commutative ring B and X is indeterminate, which is of independent interest, and along with several characterization theorems, we prove that in such a ring, each proper ideal is a finite product of n-absorbing ideals for some n ≥ 2 if and only if A ⊆ B is essentially a finite product of field extensions. A complete description of when an order of a quadratic number field is a locally pseudo valuation domain, a locally almost pseudo valuation domain or a locally conducive domain is given.

A note on the Diophantine equations [formula omitted] and application to odd perfect numbers

Let N be an odd perfect number. Then, Euler proved that there exist some integers n,α and a prime q such that N=n2qα, q∤n, and q≡α≡1mod4. In this note, we prove that the ratio σ(n2)qα is neither a square nor a square times a single prime unless α=1. It is a direct consequence of a certain property of the Diophantine equation 2ln2=1+q+⋯+qα, where l denotes one or a prime, and its proof is based on the prime ideal factorization in the quadratic orders Z[1−q] and the primitive solutions of generalized Fermat equations xβ+yβ=2z2. We give also a slight generalization to odd multiply perfect numbers.

A Generalization of Eisenstein-Schönemann’s Irreducibility Criterion

Abstract The Eisenstein criterion is a particular case of the Schönemann’s irreducibility criterion stated in 1846. In 1906, based on Newton polygon techniques, Dumas gave a generalization of the Eisenstein criterion. In this paper, we extend this last generalization. Some applications on factorization of polynomials, and prime ideal factorization will be given, too.

Algorithms for Quadratic Forms over Global Function Fields

This dissertation presents computational tools for quadratic forms over global function fields of characteristic different from 2. The majority of the algorithms we develop in this study rely on our ability to find all places dividing any coefficient of the given quadratic form. This problem is equivalent to the factorization of fractional ideals in the ring of polynomial functions of a global function field. As a result, we begin by presenting alternative approaches for factoring fractional ideals that do not rely on determining the maximum order of the global function field in question. We then propose techniques for tackling the following quadratic form theory computational problems: how to detect whether a quadratic form is isotropic or not, how to detect whether a quadratic form is hyperbolic or not, how to compute the anisotropic dimension (or equivalently the Witt index) of a quadratic form, how to construct an anisotropic part of a quadratic form, how to determine if two forms are Witt-similar or are Ono-similar or not. We further explore algorithms for computing some important field invariants that are linked to quadratic forms. Those are: the length of a sum of squares, the level of a field, the Pythagoras number of a field, as well as a Pythagoras element of a field.

Algorithms for Quadratic Forms over Global Function Fields

This dissertation presents computational tools for quadratic forms over global function fields of characteristic different from 2. The majority of the algorithms we develop in this study rely on our ability to find all places dividing any coefficient of the given quadratic form. This problem is equivalent to the factorization of fractional ideals in the ring of polynomial functions of a global function field. As a result, we begin by presenting alternative approaches for factoring fractional ideals that do not rely on determining the maximum order of the global function field in question. We then propose techniques for tackling the following quadratic form theory computational problems: how to detect whether a quadratic form is isotropic or not, how to detect whether a quadratic form is hyperbolic or not, how to compute the anisotropic dimension (or equivalently the Witt index) of a quadratic form, how to construct an anisotropic part of a quadratic form, how to determine if two forms are Witt-similar or are Ono-similar or not. We further explore algorithms for computing some important field invariants that are linked to quadratic forms. Those are: the length of a sum of squares, the level of a field, the Pythagoras number of a field, as well as a Pythagoras element of a field.

On power integral bases for certain pure sextic fields

In their paper [1], Shahzad Ahmad et al. given a characterization on any pure sextic number field Q(m1/6) with square-free integers m satisfying m 6 ±1 (mod 9) to have a power integral bases or do not. In this paper, for these results, we give a new easier proof than that given in [1]. We further investigate the cases m 1 (mod 4) independently to the satisfaction of m2 1 (mod 9), m 1 (mod 9), and the number fields defined by x2r3t−m, where r, t are two non-negative integers, and m is a square free integer are investigated. The proposed proofs are based on Dedekind’s criterion and on prime ideal factorization.

Ideal factorization in strongly discrete independent rings of Krull type

Let D be an integral domain and w be the so-called w-operation on D. In this paper, we define the notion of w-ZPUI domains which is a natural generalization of ZPUI domains introduced by Olberding in 2000. We say that D is a w-ZPUI domain if every nonzero proper w-ideal I of D can be written as [Formula: see text] for some w-invertible ideal J of D and [Formula: see text] is a nonempty collection of pairwise w-comaximal prime w-ideals of D. Then, among other things, we show that D is a w-ZPUI domain if and only if the polynomial ring [Formula: see text] is a w-ZPUI domain, if and only if D is a strongly discrete independent ring of Krull type. We construct three types of new w-ZPUI domains from an old one by [Formula: see text]-construction, pullback, and [Formula: see text]-domains. We also show that given an abelian group G, there is a ZPUI domain with ideal class group G but not a Dedekind domain. Finally, we introduce and study the notion of w-ISP domains as a generalization of w-ZPUI domains.

On infinite extensions of Dedekind domains, upper semicontinuous functions and the ideal class semigroups

We study the monoid of fractional ideals and the ideal class semigroup of an arbitrary given one dimensional normal domain 𝔒 obtained by an infinite integral extension of a Dedekind domain. We introduce a notion of “upper semicontinuous functions” whose domain is the maximal spectrum of 𝔒 equipped with the inverse topology introduced by Hochster, and whose codomain is a certain totally ordered monoid containing ℝ. We construct an isomorphism between a monoid consisting of such upper semicontinuous functions satisfying certain conditions and the monoid of fractional ideals of 𝔒. This result can be regarded as a generalization of the theory of prime ideal factorization for Dedekind domains. By using such isomorphism, we study the Galois-monoid structure of the ideal class semigroup of 𝔒.

Variations of primeness and factorization of ideals in Leavitt path algebras

In this paper we describe three different variations of prime ideals: strongly irreducible ideals, strongly prime ideals and insulated prime ideals in the context of Leavitt path algebras. We give necessary and sufficient conditions under which a proper ideal of a Leavitt path algebra L is a product as well as an intersection of finitely many of these different types of prime ideals. Such factorizations, when they are irredundant, are shown to be unique except for the order of the factors. We also characterize the Leavitt path algebras L in which every ideal admits such factorizations and also in which every ideal is one of these special type of ideals.

Unique factorization of ideals in commutative rings with zero divisors

“Unique factorization” was central to the initial development of ideal theory. We update this topic with several new results concerning notions of “unique ideal factorization rings” with zero divisors. Along the way, we obtain new characterizations of several well-known kinds of rings in terms of their ideal factorization properties and examine when monoid rings satisfy various kinds of “unique ideal factorization.” Our results include necessary and sufficient conditions for a monoid ring with S cancellative to be a π-ring, a higher-dimensional generalization of Hardy and Shores’s classic characterization of when is a general Zerlegung Primideale ring.

Factorization of polynomials over valued fields based on graded polynomials

Abstract In this paper, we develop a new method based on Newton polygon and graded polynomials, similar to the known one based on Newton polygon and residual polynomials. This new method allows us the factorization of any monic polynomial in any henselian valued field. As applications, we give a new proof of Hensel’s lemma and a theorem on prime ideal factorization.

Index, the prime ideal factorization in simplest quartic fields and counting their discriminants

We consider the simplest quartic number fields Km defined by the irreducible quartic polynomials x4-mx3-6x2+mx+1, where m runs over the positive rational integers such that the odd part of m2+16 is square free. In this paper, we study the index I(Km) and determine the explicit prime ideal factorization of rational primes in simplest quartic number fields Km. On the other hand, we establish an asymptotic formula for the number of simplest quartic fields with discriminant ? x and given index.

Ideal structure and factorization properties of the regular kernel operators

We consider the structure of the lattice of (order and algebra) ideals of the band of regular kernel operators on L^p-spaces. We show, in particular, that for any L^p(mu ) space, with mu sigma -finite and 1<p<infty , the norm-closure of the ideal of finite-rank operators on L^p(mu ), is the only non-trivial proper closed (order and algebra) ideal of this band. Key to our results in the L^p setting is the fact that every regular kernel operator on an L^p(mu ) space (mu and p as before) factors with regular factors through ell _p. We show that a similar but weaker factorization property, where ell _p is replaced by some reflexive purely atomic Banach lattice, characterizes the regular kernel operators from a reflexive Banach lattice with weak order unit to a KB-space with weak order unit.

Intrinsic factorization of ideals in Dedekind domains

We present a generalization of a polynomial factorization algorithm that works with ideals in maximal orders of global function fields. The method presented in this paper is intrinsic in the sense that it does not depend on the embedding of the ring of polynomials into the Dedekind domain in question.

Intrinsic Factorization of Ideals in Dedekind Domains

We present a generalization of a polynomial factorization algorithm that works with ideals in maximal orders of global function fields. The method presented in this paper is intrinsic in the sense that it does not depend on the embedding of the ring of polynomials into the Dedekind domain in questi on.

Factorization of ideals

We study the factorization of ideals of a commutative ring, defining multiple different kinds of “nonfactorable” ideals and several “factorability” properties weaker than unique factorization. We characterize (some of) these notions, determine the implications between them, and give several examples to illustrate the differences. We also examine how these properties behave with respect to localization, direct products, idealizations, polynomial rings, monoid domains, (generalized) power series rings, and the classical D + M construction. Along the way, we give some new characterizations of the finite superideal rings introduced by A.J. Hetzel and A.M. Lawson.

Prime ideal factorization and p-integral basis of quintic number fields defined by x^5+ax+b

Based on Newton polygon techniques, for every prime integer p, a p-integral basis of ℤK, and the factorization of the principal ideal pℤK into prime ideals of ℤK are given, where K is a quintic number field defined by an irreducible trinomial X5 + aX + b ∈ ℤ[X].

U-factorization of ideals

We study the factorization of ideals of a commutative ring, in the context of the U-factorization framework introduced by Fletcher. This leads to several “U-factorability” properties weaker than unique U-factorization. We characterize these notions, determine the implications between them, and give several examples to illustrate the differences. For example, we show that a ring is a general ZPI-ring if and only if its monoid of ideals has unique factorization in the sense of Fletcher. We also examine how these “U-factorability” properties behave with respect to several ring-theoretic constructions.

CM values of regularized theta lifts and harmonic weak Maaß forms of weight 1

We study special values of regularized theta lifts at complex multiplication (CM) points. In particular, we show that CM values of Borcherds products can be expressed in terms of finitely many Fourier coefficients of certain harmonic weak Maa{\ss} forms of weight one. As it turns out, these coefficients are logarithms of algebraic integers whose prime ideal factorization is determined by special cycles on an arithmetic curve. Our results imply a conjecture of Duke and Li and give a new proof of the modularity of a certain arithmetic generating series of weight one studied by Kudla, Rapoport and Yang. The results of the paper are much improved in comparison to the 2012 preprint arxiv:1208.2386 which contained partial results in the same direction. Moreover, they are also an improvement of the main result of the authors thesis (CM values of regularized theta lifts, TU Darmstadt, 2013).

Prime ideal factorization in quartic number fields

For every prime integer p, and for every number field K defined by a p-regular polynomial, the form of the factorization of the principal ideal pℤK into prime ideals of ℤK is given. To illustrate the potential applications of this factorization, we derive from this result an explicit description of the factorization of pℤK, where K is a quartic number field defined by an irreducible polynomial X4 + aX + b ∈ ℤ[X].