scholarly journals Note on soft fractional ideal of ring

2018 ◽  
Keyword(s):  
Fractional Ideal

THE v-OPERATION IN EXTENSIONS OF INTEGRAL DOMAINS

2012 ◽  
Vol 11 (01) ◽  
pp. 1250007 ◽  
Author(s):  
DAVID F. ANDERSON ◽  
SAID EL BAGHDADI ◽  
MUHAMMAD ZAFRULLAH
Keyword(s):  
Finitely Generated ◽  
Integral Domains ◽  
Fractional Ideal

An extension D ⊆ R of integral domains is strongly t-compatible (respectively, t-compatible) if (IR)-1 = (I-1R)v (respectively, (IR)v = (IvR)v) for every nonzero finitely generated fractional ideal I of D. We show that strongly t-compatible implies t-compatible and give examples to show that the converse does not hold. We also indicate situations where strong t-compatibility and its variants show up naturally. In addition, we study integral domains D such that D ⊆ R is strongly t-compatible (respectively, t-compatible) for every overring R of D.

scholarly journals Integral ∨-ideals

1981 ◽  
Vol 22 (2) ◽  
pp. 167-172 ◽  
Author(s):  
David F. Anderson
Keyword(s):  
Maximal Ideal ◽  
Integral Domain ◽  
Dedekind Domain ◽  
Integral Ideal ◽  
Quotient Field ◽  
Fractional Ideal

Let R be an integral domain with quotient field K. A fractional ideal I of R is a ∨-ideal if I is the intersection of all the principal fractional ideals of R which contain I. If I is an integral ∨-ideal, at first one is tempted to think that I is actually just the intersection of the principal integral ideals which contain I.However, this is not true. For example, if R is a Dedekind domain, then all integral ideals are ∨-ideals. Thus a maximal ideal of R is an intersection of principal integral ideals if and only if it is actually principal. Hence, if R is a Dedekind domain, each integral ∨-ideal is an intersection of principal integral ideals precisely when R is a PID.

scholarly journals Normal Bases in Galois Extensions of Number Fields

1969 ◽  
Vol 34 ◽  
pp. 153-167 ◽  
Author(s):  
S. Ullom
Keyword(s):  
Number Field ◽  
Number Fields ◽  
Abstract Algebra ◽  
Normal Basis ◽  
Galois Module ◽  
Normal Bases ◽  
Ring Of Integers ◽  
Root Numbers ◽  
Fractional Ideal ◽  
Rings Of Integers

The notion of module together with many other concepts in abstract algebra we owe to Dedekind [2]. He recognized that the ring of integers OK of a number field was a free Z-module. When the extension K/F is Galois, it is known that K has an algebraic normal basis over F. A fractional ideal of K is a Galois module if and only if it is an ambiguous ideal. Hilbert [4, §§105-112] used the existence of a normal basis for certain rings of integers to develop the theory of root numbers — their decomposition already having been studied by Kummer.

A product theorem for ideals over orders

1970 ◽  
Vol 68 (1) ◽  
pp. 17-20 ◽  
Author(s):  
Michael Singer
Keyword(s):  
Dedekind Domain ◽  
Product Theorem ◽  
Fractional Ideal ◽  
Dedekind Domains

Fröhlich(1) has obtained certain invariants for modules over commutative separable orders over dedekind domains, whose values are ideals in the dedekind domain. In two particular applications he has shown that these invariants supply a criterion for a module over a given order to be projective ((1), Theorem 4), and another one for a fractional ideal to be invertible ((1), Theorem 5).

scholarly journals Note on Colon-Multiplication Domains

2010 ◽  
Vol 2010 ◽  
pp. 1-4 ◽  
Author(s):  
A. Mimouni
Keyword(s):  
Maximal Ideal ◽  
Integral Domain ◽  
Nonzero Ideal ◽  
Quotient Field ◽  
Integral Domains ◽  
Fractional Ideal ◽  
Multiplication Ideal

LetRbe an integral domain with quotient fieldL.Call a nonzero (fractional) idealAofRa colon-multiplication ideal any idealA, such thatB(A:B)=Afor every nonzero (fractional) idealBofR.In this note, we characterize integral domains for which every maximal ideal (resp., every nonzero ideal) is a colon-multiplication ideal. It turns that this notion unifies Dedekind andMTPdomains.

scholarly journals On error distributions in ring-based LWE

2016 ◽  
Vol 19 (A) ◽  
pp. 130-145 ◽  
Author(s):  
Wouter Castryck ◽  
Ilia Iliashenko ◽  
Frederik Vercauteren
Keyword(s):  
Recent Literature ◽  
Number Fields ◽  
Scaling Up ◽  
Cryptographic Primitives ◽  
Fractional Ideal ◽  
Error Distributions ◽  
Secret Keys ◽  
Learning With Errors ◽  
Learning With Errors Problem ◽  
Made In

Since its introduction in 2010 by Lyubashevsky, Peikert and Regev, the ring learning with errors problem (ring-LWE) has become a popular building block for cryptographic primitives, due to its great versatility and its hardness proof consisting of a (quantum) reduction from ideal lattice problems. But, for a given modulus$q$and degree$n$number field$K$, generating ring-LWE samples can be perceived as cumbersome, because the secret keys have to be taken from the reduction mod$q$of a certain fractional ideal${\mathcal{O}}_{K}^{\vee }\subset K$called the codifferent or ‘dual’, rather than from the ring of integers${\mathcal{O}}_{K}$itself. This has led to various non-dual variants of ring-LWE, in which one compensates for the non-duality by scaling up the errors. We give a comparison of these versions, and revisit some unfortunate choices that have been made in the recent literature, one of which is scaling up by${|\unicode[STIX]{x1D6E5}_{K}|}^{1/2n}$with$\unicode[STIX]{x1D6E5}_{K}$the discriminant of$K$. As a main result, we provide, for any$\unicode[STIX]{x1D700}>0$, a family of number fields$K$for which this variant of ring-LWE can be broken easily as soon as the errors are scaled up by${|\unicode[STIX]{x1D6E5}_{K}|}^{(1-\unicode[STIX]{x1D700})/n}$.

Hecke L-Functions and the Distribution of Totally Positive Integers

2007 ◽  
Vol 59 (4) ◽  
pp. 673-695 ◽  
Author(s):  
Avner Ash ◽  
Solomon Friedberg
Keyword(s):  
Real Number ◽  
Number Field ◽  
Expected Value ◽  
Fractional Ideal ◽  
Compact Torus ◽  
Positive Elements ◽  
Totally Positive ◽  
Totally Real ◽  
Positive Integers ◽  
Real Number Field

AbstractLet K be a totally real number field of degree n. We show that the number of totally positive integers (or more generally the number of totally positive elements of a given fractional ideal) of given trace is evenly distributed around its expected value, which is obtained fromgeometric considerations. This result depends on unfolding an integral over a compact torus.

ON ⋆-COLON-MULTIPLICATION DOMAINS

2012 ◽  
Vol 12 (02) ◽  
pp. 1250156
Author(s):  
OLIVIER A. HEUBO-KWEGNA
Keyword(s):  
Maximal Ideal ◽  
Integral Domain ◽  
Fractional Ideal ◽  
Star Operation ◽  
Multiplication Ideal

Let ⋆ be a star operation on an integral domain R. An ideal A is a ⋆-colon-multiplication ideal if A⋆ = (B(A : B))⋆ for all fractional ideal B of R. We prove that every maximal ideal of R is a ⋆-colon-multiplication ideal if and only if R is a ⋆-CICD or R is a local ⋆-MTP domain. It is also shown that every ideal of R is ⋆-colon-multiplication if and only if R is a ⋆-CICD.

Sign in / Sign up

Export Citation Format

Share Document