On the Weak Global Dimension of Pseudovaluation Domains

1978 ◽  
Vol 21 (2) ◽  
pp. 159-164 ◽  
Author(s):  
David E. Dobbs
Keyword(s):  
Integral Domain ◽  
Global Dimension ◽  
Valuation Domain ◽  
Valuation Domains ◽  
Dimension 2 ◽  
The Ideal

In [7], Hedstrom and Houston introduce a type of quasilocal integral domain, therein dubbed a pseudo-valuation domain (for short, a PVD), which possesses many of the ideal-theoretic properties of valuation domains. For the reader′s convenience and reference purposes, Proposition 2.1 lists some of the ideal-theoretic characterizations of PVD′s given in [7]. As the terminology suggests, any valuation domain is a PVD. Since valuation domains may be characterized as the quasilocal domains of weak global dimension at most 1, a homological study of PVD's seems appropriate. This note initiates such a study by establishing (see Theorem 2.3) that the only possible weak global dimensions of a PVD are 0, 1, 2 and ∞. One upshot (Corollary 3.4) is that a coherent PVD cannot have weak global dimension 2: hence, none of the domains of weak global dimension 2 which appear in [10, Section 5.5] can be a PVD.

scholarly journals Anti-integral extensions $ {R[{\alpha}]/R$

2004 ◽  
Vol 35 (1) ◽  
pp. 1-12 ◽  
Author(s):  
Kiyoshi Baba ◽  
Ken-Ichi Yoshida
Keyword(s):  
Positive Integer ◽  
Integral Domain ◽  
Arbitrary Positive Integer ◽  
Integral Element ◽  
The Ideal

Let $ R $ be an integral domain and $ \alpha $ an anti-integral element of degree $ d $ over $ R $. In the paper [3] we give a condition for $ \alpha^2-a$ to be a unit of $ R[\alpha] $. In this paper we will generalize the result to an arbitrary positive integer $n$ and give a condition, in terms of the ideal $ I_{[\alpha]}^{n}D(\sqrt[n]{a}) $ of $ R $, for $ \alpha^{n}-a$ to be a unit of $ R[\alpha] $.

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.

scholarly journals On complete integral closure and Archimedean valuation domains

1996 ◽  
Vol 61 (3) ◽  
pp. 377-380 ◽  
Author(s):  
Robert Gilmer
Keyword(s):  
Integral Domain ◽  
Integral Closure ◽  
Extension Field ◽  
Quotient Field ◽  
Valuation Domains ◽  
Complete Integral Closure

AbstractSuppose D is an integral domain with quotient field K and that L is an extension field of K. We show in Theorem 4 that if the complete integral closure of D is an intersection of Archimedean valuation domains on K, then the complete integral closure of D in L is an intersection of Archimedean valuation domains on L; this answers a question raised by Gilmer and Heinzer in 1965.

A Localization of R[x]

1981 ◽  
Vol 33 (1) ◽  
pp. 103-115 ◽  
Author(s):  
James A. Huckaba ◽  
Ira J. Papick
Keyword(s):  
Integral Domain ◽  
Closed Sets ◽  
Interesting Paper ◽  
Object Of Study ◽  
The Ideal

Throughout this paper, R will be a commutative integral domain with identity and x an indeterminate. If ƒ ∈ R[x], let CR(ƒ) denote the ideal of R generated by the coefficients of ƒ. Define SR = {ƒ ∈ R[x]: cR(ƒ) = R} and UR = {ƒ ∈ R(x): cR(ƒ)– 1 = R}. For a,b ∈ R, write . When no confusion may result, we will write c(ƒ), S, U, and (a:b). It follows that both S and U are multiplicatively closed sets in R[x] [7, Proposition 33.1], [17, Theorem F], and that R[x]s ⊆ R[x]U.The ring R[x]s, denoted by R(x), has been the object of study of several authors (see for example [1], [2], [3], [12]). An especially interesting paper concerning R(x) is that of Arnold's [3], where he, among other things, characterizes when R(x) is a Priifer domain. We shall make special use of his results in our work.

scholarly journals On 2-absorbing ideals of commutative rings

2007 ◽  
Vol 75 (3) ◽  
pp. 417-429 ◽  
Author(s):  
Ayman Badawi
Keyword(s):  
Prime Ideal ◽  
Maximal Ideal ◽  
Integral Domain ◽  
Commutative Rings ◽  
Dedekind Domain ◽  
Proper Ideal ◽  
Prime Ideals ◽  
Maximal Ideals ◽  
Noetherian Domain ◽  
Valuation Domains

Suppose that R is a commutative ring with 1 ≠ 0. In this paper, we introduce the concept of 2-absorbing ideal which is a generalisation of prime ideal. A nonzero proper ideal I of R is called a 2-absorbing ideal of R if whenever a, b, c ∈ R and abc ∈ I, then ab ∈ I or ac ∈ I or bc ∈ I. It is shown that a nonzero proper ideal I of R is a 2-absorbing ideal if and only if whenever I1I2I3 ⊆ I for some ideals I1,I2,I3 of R, then I1I2 ⊆ I or I2I3 ⊆ I or I1I3 ⊆ I. It is shown that if I is a 2-absorbing ideal of R, then either Rad(I) is a prime ideal of R or Rad(I) = P1 ⋂ P2 where P1,P2 are the only distinct prime ideals of R that are minimal over I. Rings with the property that every nonzero proper ideal is a 2-absorbing ideal are characterised. All 2-absorbing ideals of valuation domains and Prüfer domains are completely described. It is shown that a Noetherian domain R is a Dedekind domain if and only if a 2-absorbing ideal of R is either a maximal ideal of R or M2 for some maximal ideal M of R or M1M2 where M1,M2 are some maximal ideals of R. If RM is Noetherian for each maximal ideal M of R, then it is shown that an integral domain R is an almost Dedekind domain if and only if a 2-absorbing ideal of R is either a maximal ideal of R or M2 for some maximal ideal M of R or M1M2 where M1,M2 are some maximal ideals of R.

The ideal transform and overrings of an integral domain

1968 ◽  
Vol 107 (4) ◽  
pp. 301-306 ◽  
Author(s):  
Jim Brewer
Keyword(s):  
Integral Domain ◽  
The Ideal

On t- Spec(R[[X]])

1995 ◽  
Vol 38 (2) ◽  
pp. 187-195 ◽  
Author(s):  
David E. Dobbs ◽  
Evan G. Houston
Keyword(s):  
Maximal Ideal ◽  
Integral Domain ◽  
Valuation Domain ◽  
Finitely Generated ◽  
Finite Dimensional ◽  
Star Operation

AbstractLet D be an integral domain, and let X be an analytic indeterminate. As usual, if I is an ideal of D, set It = ∪{JV = (J-1)-1 | J is a nonzero finitely generated subideal of I}; this defines the t-operation, a particularly useful star-operation on D. We discuss the t-operation on R[[X]], paying particular attention to the relation between t- dim(R) and t- dim(R[[X]]). We show that if P is a t-prime of R, then P[[X]] contains a t-prime which contracts to P in R, and we note that this does not quite suffice to show that t- dim(R[[X]]) ≥ t- dim(R) in general. If R is Noetherian, it is easy to see that t- dim(R[[X]]) ≥ t- dim(R), and we show that we have equality in the case of t-dimension 1. We also observe that if V is a valuation domain, then t-dim(V[[X]]) ≥ t- dim(V), and we give examples to show that the inequality can be strict. Finally, we prove that if V is a finite-dimensional valuation domain with maximal ideal M, then MV[[X]] is a maximal t-ideal of V[[X]].

scholarly journals On the automorphisms of the group ring of a unique product group

1987 ◽  
Vol 30 (2) ◽  
pp. 201-205 ◽  
Author(s):  
A. A. Mehrvarz ◽  
D. A. R. Wallace
Keyword(s):  
Group Ring ◽  
Integral Domain ◽  
Nilpotent Ideal ◽  
Product Group ◽  
Unique Product ◽  
Group 6 ◽  
Do So ◽  
The Ideal

Let R be a ring with an identity and a nilpotent ideal N. Let G be a group and let R(G) be the group ring of G over R. The aim of this paper is to study the relationships between the automorphisms of G and R-linear automorphisms of R(G) which either preserve the augmentation or do so modulo the ideal N. We shall show, for example, that if G is a unique product group ([6], Chapter 13, Section 1) then every automorphism of R(G) is modulo N induced from some automorphism of G. This result, which is immediate if, for instance, R is an integral domain, is here requiring of proof since R(G) has non-trivial units (e.g. if N ≠ 0, 1 + n(g − h), ∀ n ∈ N, ∀g, h ∈ G is a unit of augmentation 1), the existence of which is responsible for some of the difficulties inherent in the present investigation. We are obliged to the referee for several helpful suggestions and, in particular, for the proof of Lemma 2.2 whose use obviates our previous combinatorial arguments.

scholarly journals On the Factorization of Polynomials Over Discrete Valuation Domains

2014 ◽  
Vol 22 (1) ◽  
pp. 273-280
Author(s):  
Doru Ştefănescu
Keyword(s):  
Valuation Domain ◽  
Discrete Valuation Domain ◽  
Valuation Domains ◽  
Univariate Polynomials ◽  
Factorization Of Polynomials

AbstractWe study some factorization properties for univariate polynomials with coefficients in a discrete valuation domain (A,v). We use some properties of the Newton index of a polynomial to deduce conditions on v(ai) that allow us to find some information on the degree of the factors of F.

QF - 1 Rings of Global Dimension ≦ 2

1973 ◽  
Vol 25 (2) ◽  
pp. 345-352
Author(s):  
Claus Michael Ringel
Keyword(s):  
Global Dimension ◽  
The Other ◽  
Frobenius Rings ◽  
Other Hand ◽  
Dimension 2 ◽  
Left And Right

R. M. Thrall [10] introduced QF — 1, QF — 2 and QF — 3 rings as generalizations of quasi-Frobenius rings. (For definitions, see section 1. It should be noted that all rings considered are assumed to be left and right artinian.) He proved that QF — 2 rings are QF — 3 and asked whether all QF — 1 rings are QF — 2, or, at least, QF — 3. In [9] we have shown that QF — 1 rings are very similar to QF — 3 rings. On the other hand, K. Morita [6] gave two examples of QF — 1 rings, one of them not QF — 2 and therefore not QF — 3, the other one QF — 3, but not QF — 2.

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