Fractional Ideals Research Articles

Standard Bases for fractional ideals of the local ring of an algebroid curve

In this paper we present an algorithm to compute a Standard Basis for a fractional ideal I of the local ring O of an n-space algebroid curve with several branches. This allows us to determine the semimodule of values of I. When I=O, we may obtain a (finite) set of generators of the semiring of values of the curve, which determines its classical semigroup. In the complex context, identifying the Kähler differential module ΩO/C of a plane curve with a fractional ideal of O and applying our algorithm, we can compute the set of values of ΩO/C, which is an important analytic invariant associated to the curve.

The group of invertible ideals of a Prüfer ring

Let R be a commutative ring and $$\mathcal {I}(R)$$ denote the multiplicative group of all invertible fractional ideals of R, ordered by $$A \leqslant B$$ if and only if $$B \subseteq A$$ . We investigate when there is an order homomorphism from $$\mathcal {I}(R)$$ into the cardinal direct sum $$\coprod _{i \in I} G_i$$ , where $$G_i$$ ’s are value groups, if R is a Marot Prufer ring of finite character. Furthermore, over Prufer rings with zero-divisors, we investigate the conditions that make this monomorphism onto.

On the Colength of Fractional Ideals

The main result in this paper is to supply a recursive formula, on the number of minimal primes, for the colength of a fractional ideal in terms of the maximal points of the value set of the ideal itself. The fractional ideals are taken in the class of complete admissible rings, a more general class of rings than those of algebroid curves. For such rings with two or three minimal primes, a closed formula for that colength is provided, so improving results by Barucci, D'Anna and Fr\oberg.

Computing abelian varieties over finite fields isogenous to a power

In this paper we give a module-theoretic description of the isomorphism classes of abelian varieties A isogenous to B^r, where the characteristic polynomial g of Frobenius of B is an ordinary square-free q-Weil polynomial, for a power q of a prime p, or a square-free p-Weil polynomial with no real roots. Under some extra assumptions on the polynomial g we give an explicit description of all the isomorphism classes which can be computed in terms of fractional ideals of an order in a finite product of number fields. In the ordinary case, we also give a module-theoretic description of the polarizations of A.

Improved Independent Set Conditions for Fractional Factors

A graph G is called a fractional (g, f,n1, m)-critical deleted graph if after deleting any n1 vertices from G, the resulting graph admits a fractional (g, f, m)-deleted graph. A graph G is called a fractional ID-(g, f, m)-deleted if after deleting any independent set I from G, the resulting graph admits a fractional (g, f, m)-deleted graph. In this paper, we improve independent set conditions for a graph to be fractional (g, f,n1,m)-critical deleted and fractional ID-(g, f, m)-deleted. Furthermore, we present some examples to show the sharpness of given independent set bounds.

Parameters and fractional factors in different settings

In computer networks, the feasibility of data transmission can be measured in terms of fractional factor model, and specifically the framework of fractional critical deleted graph can express the setting when certain sites and channels are unavailable at a special time. We study the relationship between some parameters in graphs and the existence of fractional (g,f)-factor in various settings here. Our main contributions are three-fold: first, a connectivity condition for a graph to be fractional (g,f,n',m)-critical deleted is determined; second, the relationship between independence number and fractional ID-(g,f,m)-deleted graphs is studied; third, an isolated toughness bound for fractional (g,f)-factors is given in balanced bipartite graph setting. Furthermore, by showing counterexamples we explain that bounds for parameters are tight in some sense, and corresponding conditions in all fractional factor settings are discussed as well. Finally, two open problems are proposed for future research.

On the class semigroup of the cyclotomic $\mathbb Z_p$-extension of the rational numbers

For a commutative integral domain, the class semigroup and the class group are defined as the quotient of the semigroup of fractional ideals and the group of invertible ideals by the group of principal ideals, respectively. Let $p$ be a prime number. In algebraic number theory, especially in Iwasawa theory, the class group of the ring of integers $\mathcal{O} $ of the cyclotomic $\mathbb {Z}_{p}$-extension of the rational numbers has been studied for a long time. However, the class semigroup of $\mathcal{O} $ is not well known. We are interested in the structure of the class semigroup of $\mathcal{O} $. In order to study it, we focus on the structure of the complement set of the class group in the class semigroup of $\mathcal{O} $. In this paper, we prove that the complement set is a group and determine its structure.

Duality on value semigroups

We establish a combinatorial counterpart of the Cohen-Macaulay duality on a class of curve singularities which includes algebroid curves. For such singularities the value semigroup and the value semigroup ideals of all fractional ideals satisfy axioms that define so-called good semigroups and good semigroup ideals. We prove that each good semigroup admits a canonical good semigroup ideal which gives rise to a duality on good semigroup ideals. We show that the Cohen-Macaulay duality and our good semigroup duality are compatible under taking values.

Tight independent set neighborhood union condition for fractional critical deleted graphs and ID deleted graphs

The problem of data transmission in communication network can betransformed into the problem of fractional factor existing in graph theory. Inrecent years, the data transmission problem in the specificnetwork conditions has received a great deal of attention, and itraises new demands to the corresponding mathematical model. Underthis background, many advanced results are presented on fractionalcritical deleted graphs and fractional ID deleted graphs. In thispaper, we determine that $G$ is a fractional$ (g,f,n',m) $-critical deleted graph if$ δ(G)≥\frac{b^{2}(i-1)}{a}+n'+2m $, $ n>\frac{(a+b)(i(a+b)+2m-2)+bn'}{a} $, and \begin{document}$|N_{G}(x_{1})\cup N_{G}(x_{2})\cup···\cup N_{G}(x_{i})|≥\frac{b(n+n')}{a+b}$ \end{document} for any independent subset $ \{x_{1},x_{2},..., x_{i}\} $ of $ V(G) $. Furthermore, the independent set neighborhood union condition for a graph to be fractional ID-$ (g,f,m) $-deleted is raised. Some examples will be manifested to show the sharpness of independent set neighborhood union conditions.

A Neighborhood Union Condition for Fractional ID-[a, b]-factor-critical Graphs

Let a, b, r be nonnegative integers with $$1\leq{a}\leq{b}$$ and $$r\geq2$$ . Let G be a graph of order n with $$n >\frac{(a+2b)(r(a+b)-2)}{b}$$ . In this paper, we prove that G is fractional ID-[a, b]-factor-critical if $$\delta(G)\geq\frac{bn}{a+2b}+a(r-1)$$ and $$\mid N_{G}(x_{1}) \cup N_{G}(x_{2}) \cup \cdotp \cdotp \cdotp \cup N_{G}(x_{r})\mid\geq\frac{(a+b)n}{a+2b}$$ for any independent subset {x1, x2, · · ·, xr} in G. It is a generalization of Zhou et al.’s previous result [Discussiones Mathematicae Graph Theory, 36: 409–418 (2016)] in which r = 2 is discussed. Furthermore, we show that this result is best possible in some sense.

Good subsemigroups of ℕn

Value semigroups of non-irreducible singular algebraic curves and their fractional ideals are submonoids of [Formula: see text] that are closed under infimums, have a conductor and fulfill a special compatibility property on their elements. Monoids of [Formula: see text] fulfilling these three conditions are known in the literature as good semigroups and there are examples of good semigroups that are not realizable as the value semigroup of an algebraic curve. In this paper, we consider good semigroups independently from their algebraic counterpart, in a purely combinatorial setting. We define the concept of good system of generators, and we show that minimal good systems of generators are unique. Moreover, we give a constructive way to compute the canonical ideal and the Arf closure of a good subsemigroup when [Formula: see text].

Commutative Hopf–Galois module structure of tame extensions

We prove three theorems concerning the Hopf–Galois module structure of fractional ideals in a finite tamely ramified extension of p-adic fields or number fields which is H-Galois for a commutative Hopf algebra H. Firstly, we show that if L/K is a tame Galois extension of p-adic fields then each fractional ideal of L is free over its associated order in H. We also show that this conclusion remains valid if L/K is merely almost classically Galois. Finally, we show that if L/K is an abelian extension of number fields then every ambiguous fractional ideal of L is locally free over its associated order in H.

Triangular bases of integral closures

In this work, we consider the problem of computing triangular bases of integral closures of one-dimensional local rings.Let (K,v) be a discrete valued field with valuation ring O and let m be the maximal ideal. We take f∈O[x], a monic irreducible polynomial of degree n and consider the extension L=K[x]/(f(x)) as well as OL the integral closure of O in L, which we suppose to be finitely generated as an O-module.The algorithm MaxMin, presented in this paper, computes triangular bases of fractional ideals of OL. The theoretical complexity is equivalent to current state of the art methods and in practice is almost always faster. It is also considerably faster than the routines found in standard computer algebra systems, excepting some cases involving very small field extensions.

Robust Regulation of MIMO systems: A Reformulation of the Internal Model Principle

The internal model principle is a fundamental result stating a necessary and sufficient condition for a stabilizing controller to be robustly regulating. Its classical formulation is given in terms of coprime factorizations and the largest invariant factor of the signal generator which sets unnecessary restrictions for the theory and its applicability. In this article, the internal model principle is formulated using a general factorization approach and the generators of the fractional ideals generated by the elements of the signal generator. The proposed results are related to the classical ones.

The group of divisibility of a finite character intersection of valuation rings

ABSTRACTLet R be a Prüfer domain. The group of invertible fractional ideals ℑ(R) is an lattice-ordered group (ℓ-group) with respect to the ordering defined by A≤B if and only if B⊆A. In this work, we prove that if R has a finite character and each nonzero prime ideal of R contains a minimal nonzero prime ideal, then ℑ(R) is a cardinal sum of indecomposable semilocal ℓ-groups. We examine the ℓ-groups that can be realized as the group of invertible fractional ideals of a finite character Prüfer overring of .

Neighbourhood conditions for fractional ID-[a, b]-factor-critical graphs

A graph G is fractional ID-[a, b]-factor-critical if G?I has a fractional [a, b]-factor for every independent set I of G. We extend a result of Zhou and Sun concerning fractional ID-k-factor-critical graphs.

Overrings of Prüfer v-multiplication domains

Let [Formula: see text] be an integral domain, [Formula: see text] and [Formula: see text] the set of fractional ideals of [Formula: see text]. Let [Formula: see text] a finitely generated ideal with [Formula: see text]. For a torsion-free [Formula: see text]-module [Formula: see text], define [Formula: see text] for some [Formula: see text]. Call [Formula: see text] a [Formula: see text]-module if [Formula: see text]. On [Formula: see text], the function [Formula: see text] is a star-operation of finite character. An integral ideal [Formula: see text] maximal with respect to being a proper [Formula: see text]-ideal is a prime ideal called a maximal [Formula: see text]-ideal. A torsion-free [Formula: see text]-module [Formula: see text] is called [Formula: see text]-flat, if [Formula: see text] is a flat [Formula: see text]-module for each [Formula: see text], the set of maximal [Formula: see text]-ideals of [Formula: see text]. [Formula: see text] is called a Prüfer [Formula: see text]-multiplication domain (P[Formula: see text]MD), if [Formula: see text] is a valuation ring for each [Formula: see text]. We characterize [Formula: see text]-flat modules in a manner similar to the characterization of flat modules, study them when they are rings [Formula: see text] with [Formula: see text] and characterize P[Formula: see text]MDs using them and compare our work with similar work in the literature.

Group-theoretic and topological invariants of completely integrally closed Prüfer domains

We consider the lattice-ordered groups Inv(R) and Div(R) of invertible and divisorial fractional ideals of a completely integrally closed Prüfer domain. We prove that Div(R) is the completion of the group Inv(R), and we show there is a faithfully flat extension S of R such that S is a completely integrally closed Bézout domain with Div(R)≅Inv(S). Among the class of completely integrally closed Prüfer domains, we focus on the one-dimensional Prüfer domains. This class includes Dedekind domains, the latter being the one-dimensional Prüfer domains whose maximal ideals are finitely generated. However, numerous interesting examples show that the class of one-dimensional Prüfer domains includes domains that differ quite significantly from Dedekind domains by a number of measures, both group-theoretic (involving Inv(R) and Div(R)) and topological (involving the maximal spectrum of R). We examine these invariants in connection with factorization properties of the ideals of one-dimensional Prüfer domains, putting special emphasis on the class of almost Dedekind domains, those domains for which every localization at a maximal ideal is a rank one discrete valuation domain, as well as the class of SP-domains, those domains for which every proper ideal is a product of radical ideals. For this last class of domains, we show that if in addition the ring has nonzero Jacobson radical, then the lattice-ordered groups Inv(R) and Div(R) are determined entirely by the topology of the maximal spectrum of R, and that the Cantor–Bendixson derivatives of the maximal spectrum reflect the distribution of sharp and dull maximal ideals.

A neighborhood condition for fractional ID-[a,b]-factor-critical graphs

Let $G$ be a graph of order $n$, and let $a$ and $b$ be two integers with $1\leq a\leq b$. Let $h: E(G)\rightarrow [0,1]$ be a function. If $a\leq\sum_{e\ni x}h(e)\leq b$ holds for any $x\in V(G)$, then we call $G[F_h]$ a fractional $[a,b]$-factor of $G$ with indicator function $h$ where $F_h=\{e\in E(G): h(e)>0\}$. A graph $G$ is fractional independent-set-deletable $[a,b]$-factor-critical (in short, fractional ID-$[a,b]$-factor-critical) if $G-I$ has a fractional $[a,b]$-factor for every independent set $I$ of $G$. In this paper, it is proved that if $n\geq\frac{(a+2b)(2a+2b-3)+1}{b}$, $\delta(G)\geq\frac{bn}{a+2b}+a$ and $|N_G(x)\cup N_G(y)|\geq\frac{(a+b)n}{a+2b}$ for any two nonadjacent vertices $x,y\in V(G)$, then $G$ is fractional ID-$[a,b]$-factor-critical. Furthermore, it is shown that this result is best possible in some sense.

A NEW CHARACTERIZATION OF PRÜFER v-MULTIPLICATION DOMAINS

Let $D$ be an integral domain and $w$ be the so-called $w$-operation on $D$. In this note, we introduce the notion of $*(w)$-domains: $D$ is a $*(w)$-domain if $((\cap (x_i))(\cap (y_j)))_w = \cap (x_iy_j)$ for all nonzero elements $x_1, \dots , x_m; y_1, \dots , y_n$ of $D$. We then show that $D$ is a Pr\ufer $v$-multiplication domain if and only if $D$ is a $*(w)$-domain and $A^{-1}$ is of finite type for all nonzero finitely generated fractional ideals $A$ of $D$.