On Representations of Orders Over Dedekind Domains

1960 ◽  
Vol 12 ◽  
pp. 107-125 ◽  
Author(s):  
D. G. Higman
Keyword(s):  
Integral Domain ◽  
Homological Algebra ◽  
Dedekind Domain ◽  
Main Concern ◽  
Direct Sums ◽  
Dedekind Domains ◽  
The Ideal

We study representations of o-orders, that is, of o-regular -algebras, in the case that o is a Dedekind domain. Our main concern is with those -modules, called -representation modules, which are regular as o-modules. For any -module M we denote by D(M) the ideal consisting of the elements x ∈ o such that x.Ext1(M, N) = 0 for all -modules N, where Ext = Ext(,0) is the relative functor of Hochschild (5). To compute D(M) we need the small amount of homological algebra presented in § 1. In § 2 we show that the -representation modules with rational hulls isomorphic to direct sums of right ideal components of the rational hull A of , called principal-modules, are characterized by the property that D(M) ≠ 0. The (, o)-projective -modules are those with D(M) = 0. We observe that D(M) divides the ideal I() of (2) for every M , and give another proof of the fact that I() ≠ 0 if and only if A is separable. Up to this point, o can be taken to be an arbitrary integral domain.

Factoring Ideals into Semiprime Ideals

1978 ◽  
Vol 30 (6) ◽  
pp. 1313-1318 ◽  
Author(s):  
N. H. Vaughan ◽  
R. W. Yeagy
Keyword(s):  
Integral Domain ◽  
Dedekind Domain ◽  
Prime Ideals ◽  
Semiprime Ideal ◽  
Dedekind Domains

Let D be an integral domain with 1 ≠ 0 . We consider “property SP” in D, which is that every ideal is a product of semiprime ideals. (A semiprime ideal is equal to its radical.) It is natural to consider property SP after studying Dedekind domains, which involve factoring ideals into prime ideals. We prove that a domain D with property SP is almost Dedekind, and we give an example of a nonnoetherian almost Dedekind domain with property SP.

IDEAL CHAINS IN RESIDUALLY FINITE DEDEKIND DOMAINS

2018 ◽  
Vol 99 (1) ◽  
pp. 56-67 ◽  
Author(s):  
YU-JIE WANG ◽  
YI-JING HU ◽  
CHUN-GANG JI
Keyword(s):  
Dedekind Domain ◽  
Nonzero Ideal ◽  
Counting Problems ◽  
Residually Finite ◽  
Dedekind Domains ◽  
The Ideal

Let $\mathfrak{D}$ be a residually finite Dedekind domain and let $\mathfrak{n}$ be a nonzero ideal of $\mathfrak{D}$. We consider counting problems for the ideal chains in $\mathfrak{D}/\mathfrak{n}$. By using the Cauchy–Frobenius–Burnside lemma, we also obtain some further extensions of Menon’s identity.

Primary Ideals and Prime Power Ideals

1966 ◽  
Vol 18 ◽  
pp. 1183-1195 ◽  
Author(s):  
H. S. Butts ◽  
Robert W. Gilmer
Keyword(s):  
Finite Number ◽  
Commutative Ring ◽  
Maximal Ideal ◽  
Integral Domain ◽  
Prime Power ◽  
Ideal Theory ◽  
Dedekind Domain ◽  
Primary Ideal ◽  
Primary Ideals ◽  
The Ideal

This paper is concerned with the ideal theory of a commutative ringR.We sayRhas Property (α) if each primary ideal inRis a power of its (prime) radical;Ris said to have Property (δ) provided every ideal inRis an intersection of a finite number of prime power ideals. In (2, Theorem 8, p. 33) it is shown that ifDis a Noetherian integral domain with identity and if there are no ideals properly between any maximal ideal and its square, thenDis a Dedekind domain. It follows from this that ifDhas Property (α) and is Noetherian (in which caseDhas Property (δ)), thenDis Dedekind.

A CHARACTERIZATION OF GOLDIE EXTENDING MODULES OVER DEDEKIND DOMAINS

2011 ◽  
Vol 10 (06) ◽  
pp. 1291-1299 ◽  
Author(s):  
EVRIM AKALAN ◽  
GARY F. BIRKENMEIER ◽  
ADNAN TERCAN
Keyword(s):  
Principal Ideal ◽  
Dedekind Domain ◽  
Principal Ideal Domain ◽  
Direct Sums ◽  
Ideal Domain ◽  
Dedekind Domains ◽  
Pure Submodules ◽  
Torsion Modules

In this paper, we characterize [Formula: see text]-extending (Goldie extending) modules over Dedekind domains and we use the [Formula: see text]-extending condition to characterize the modules over a principal ideal domain whose pure submodules are direct summands. Moreover, we show that if R is a principal ideal domain, then the class of [Formula: see text]-extending modules is closed under direct summands and that if R is a Dedekind domain, then the class of [Formula: see text]-extending torsion modules is closed under finite direct sums.

Primary Ideals in Prüfer Domains

1966 ◽  
Vol 18 ◽  
pp. 1024-1030 ◽  
Author(s):  
Jack Ohm
Keyword(s):  
Prime Ideal ◽  
Valuation Ring ◽  
Integral Domain ◽  
Quotient Ring ◽  
Integral Closure ◽  
Dedekind Domain ◽  
Prüfer Domain ◽  
Valuation Rings ◽  
Dedekind Domains ◽  
Primary Ideals

A Prüfer domain is an integral domainDwith the property that for every proper prime idealPofDthe quotient ringDPis a valuation ring. Examples of such domains are valuation rings and Dedekind domains, a Dedekind domain being merely a noetherian Prüfer domain. The integral closure of the integers in an infinite algebraic extension of the rationals is another example of a Prüfer domain (5, p. 555, Theorem 8). This third example has been studied initially by Krull (4) and then by Nakano (8).

INTEGRAL AND COMPLETE INTEGRAL CLOSURES OF IDEALS IN INTEGRAL DOMAINS

2011 ◽  
Vol 10 (04) ◽  
pp. 701-710
Author(s):  
A. MIMOUNI
Keyword(s):  
Integral Domain ◽  
Integral Closure ◽  
Integral Domains ◽  
Dedekind Domains ◽  
Integrally Closed ◽  
Complete Integral Closure ◽  
Integral Closures ◽  
The Ideal

This paper studies the integral and complete integral closures of an ideal in an integral domain. By definition, the integral closure of an ideal I of a domain R is the ideal given by I′ ≔ {x ∈ R | x satisfies an equation of the form xr + a1xr-1 + ⋯ + ar = 0, where ai ∈ Ii for each i ∈ {1, …, r}}, and the complete integral closure of I is the ideal Ī ≔ {x ∈ R | there exists 0 ≠ = c ∈ R such that cxn ∈ In for all n ≥ 1}. An ideal I is said to be integrally closed or complete (respectively, completely integrally closed) if I = I′ (respectively, I = Ī). We investigate the integral and complete integral closures of ideals in many different classes of integral domains and we give a new characterization of almost Dedekind domains via the complete integral closure of ideals.

Almost Multiplication Rings

1965 ◽  
Vol 17 ◽  
pp. 267-277 ◽  
Author(s):  
H. S. Butts ◽  
R. C. Phillips
Keyword(s):  
Prime Ideal ◽  
Commutative Ring ◽  
Sufficient Conditions ◽  
Unit Element ◽  
Ideal Theory ◽  
Dedekind Domain ◽  
Necessary And Sufficient Conditions ◽  
Dedekind Domains ◽  
Necessary And Sufficient ◽  
The Ideal

It is well known that an idealAin a. Dedekind domain has a prime radical if and only ifAis a power of a prime ideal. The purpose of this paper is to determine necessary and sufficient conditions in order that a commutative ring with unit element have this property and to study the ideal theory in such rings. Domains with unit element having the above property possess many of the characteristics of Dedekind domains (however, they need not be Noetherian) and will be referred to in this paper as "almost Dedekind domains"—these domains are considered in Section 1.

THE RINGS $D((\mathcal{X}))_i$ AND $D\{\{\mathcal{X}\}\}_i$

2012 ◽  
Vol 12 (02) ◽  
pp. 1250147 ◽  
Author(s):  
GYU WHAN CHANG ◽  
DONG YEOL OH
Keyword(s):  
Integral Domain ◽  
Dedekind Domain ◽  
Power Series Ring ◽  
Prüfer Domain ◽  
Series Ring ◽  
Star Operation ◽  
Krull Domain ◽  
Noetherian Domain ◽  
Infinite Set ◽  
The Ideal

Let D be an integral domain, [Formula: see text] be an infinite set of indeterminates over D, and [Formula: see text] be the i th type of power series ring over D for i = 1, 2, 3. For [Formula: see text], let c(f) denote the ideal of D generated by the coefficients of f. For a star operation * on D, put [Formula: see text], where *f is the star operation of finite type on D induced by *. Let [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text]. We show that [Formula: see text], and that D is a Noetherian domain if and only if [Formula: see text] is a Noetherian domain. We also show that D is a Krull domain if and only if [Formula: see text] is a Dedekind domain, if and only if [Formula: see text] is a Prüfer domain, and that D is a Dedekind domain if and only if [Formula: see text] is a Dedekind domain, if and only if [Formula: see text] is a Prüfer domain.

scholarly journals Nonsingular bilinear forms on direct sums of ideals

2013 ◽  
Vol 63 (4) ◽  
Author(s):  
Beata Rothkegel
Keyword(s):  
Bilinear Form ◽  
Dedekind Domain ◽  
Bilinear Forms ◽  
Direct Sums ◽  
Direct Sum

AbstractIn the paper we formulate a criterion for the nonsingularity of a bilinear form on a direct sum of finitely many invertible ideals of a domain. We classify these forms up to isometry and, in the case of a Dedekind domain, up to similarity.

scholarly journals Primitive divisors of sequences associated to elliptic curves with complex multiplication

2021 ◽  
Vol 7 (2) ◽  
Author(s):  
Matteo Verzobio
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Number Field ◽  
Complex Multiplication ◽  
Dedekind Domain ◽  
Torsion Point ◽  
Integral Ideal ◽  
Primitive Divisors ◽  
Primitive Divisor ◽  
The Ideal

AbstractLet P and Q be two points on an elliptic curve defined over a number field K. For $$\alpha \in {\text {End}}(E)$$ α ∈ End ( E ) , define $$B_\alpha $$ B α to be the $$\mathcal {O}_K$$ O K -integral ideal generated by the denominator of $$x(\alpha (P)+Q)$$ x ( α ( P ) + Q ) . Let $$\mathcal {O}$$ O be a subring of $${\text {End}}(E)$$ End ( E ) , that is a Dedekind domain. We will study the sequence $$\{B_\alpha \}_{\alpha \in \mathcal {O}}$$ { B α } α ∈ O . We will show that, for all but finitely many $$\alpha \in \mathcal {O}$$ α ∈ O , the ideal $$B_\alpha $$ B α has a primitive divisor when P is a non-torsion point and there exist two endomorphisms $$g\ne 0$$ g ≠ 0 and f so that $$f(P)= g(Q)$$ f ( P ) = g ( Q ) . This is a generalization of previous results on elliptic divisibility sequences.

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