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.

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.

On ideals preserving generalized local cohomology modules

2015 ◽  
Vol 15 (01) ◽  
pp. 1650019 ◽  
Author(s):  
Tsutomu Nakamura
Keyword(s):  
Positive Integer ◽  
Noetherian Ring ◽  
Local Cohomology ◽  
Finitely Generated ◽  
Commutative Noetherian Ring ◽  
Local Cohomology Modules ◽  
Generalized Local Cohomology Modules ◽  
Cohomology Modules ◽  
The Ideal

Let R be a commutative Noetherian ring, 𝔞 an ideal of R and M, N two finitely generated R-modules. Let t be a positive integer or ∞. We denote by Ωt the set of ideals 𝔠 such that [Formula: see text] for all i < t. First, we show that there exists the ideal 𝔟t which is the largest in Ωt and [Formula: see text]. Next, we prove that if 𝔡 is an ideal such that 𝔞 ⊆ 𝔡 ⊆ 𝔟t, then [Formula: see text] for all i < t.

scholarly journals A further result on the complex oscillation theory of periodic second order linear differential equations

1990 ◽  
Vol 33 (1) ◽  
pp. 143-158 ◽  
Author(s):  
Shian Gao
Keyword(s):  
Entire Function ◽  
Positive Integer ◽  
Oscillation Theory ◽  
Transcendental Entire Function ◽  
Arbitrary Positive Integer ◽  
Order Of Growth ◽  
Exponent Of Convergence ◽  
Complex Oscillation ◽  
Zero Sequence ◽  
Linear Differential

We prove the following: Assume that , where p is an odd positive integer, g(ζ is a transcendental entire function with order of growth less than 1, and set A(z) = B(ezz). Then for every solution , the exponent of convergence of the zero-sequence is infinite, and, in fact, the stronger conclusion holds. We also give an example to show that if the order of growth of g(ζ) equals 1 (or, in fact, equals an arbitrary positive integer), this conclusion doesn't hold.

scholarly journals A Sum Connected with Quadratic Residues

1956 ◽  
Vol 10 ◽  
pp. 1-7
Author(s):  
L. Carlitz
Keyword(s):  
Positive Integer ◽  
Legendre Symbol ◽  
Arbitrary Positive Integer ◽  
Arbitrary Integer ◽  
Quadratic Residues

Let p be a prime > 2 and m an arbitrary positive integer; definewhere (r/p) is the Legendre symbol. We consider the problem of finding the highest power of p dividing Sm. A little more generally, if we putwhere a is an arbitrary integer, we seek the highest power of p dividing Sm(a). Clearly Sm = Sm(0), and Sm(a) = Sm(b) when a ≡ b (mod p).

scholarly journals On rings all of whose factor rings are integral domains

1993 ◽  
Vol 55 (3) ◽  
pp. 325-333
Author(s):  
Yasuyuki Hirano
Keyword(s):  
Positive Integer ◽  
Zero Divisor ◽  
Arbitrary Positive Integer ◽  
Integral Domains ◽  
Factor Ring ◽  
Zero Divisors ◽  
Proper Quotient

AbstractA ring R is called a (proper) quotient no-zero-divisor ring if every (proper) nonzero factor ring of R has no zero-divisors. A characterization of a quotient no-zero-divisor ring is given. Using it, the additive groups of quotient no-zero-divisor rings are determined. In addition, for an arbitrary positive integer n, a quotient no-zero-divisor ring with exactly n proper ideals is constructed. Finally, proper quotient no-zero-divisor rings and their additive groups are classified.

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

scholarly journals A new approach to the Tarry–Escott problem

2017 ◽  
Vol 13 (02) ◽  
pp. 393-417 ◽  
Author(s):  
Ajai Choudhry
Keyword(s):  
Positive Integer ◽  
New Method ◽  
New Approach ◽  
Parametric Solutions ◽  
Ideal Solutions ◽  
The Ideal

In this paper we describe a new method of obtaining ideal solutions of the well-known Tarry–Escott problem, that is, the problem of finding two distinct sets of integers [Formula: see text] and [Formula: see text] such that [Formula: see text], [Formula: see text], where [Formula: see text] is a given positive integer. When [Formula: see text], only a limited number of parametric/numerical ideal solutions of the Tarry–Escott problem are known. In this paper, by applying the new method mentioned above, we find several new parametric ideal solutions of the problem when [Formula: see text]. The ideal solutions obtained by this new approach are more general and, very frequently, simpler than the ideal solutions obtained by the earlier methods. We also obtain new parametric solutions of certain diophantine systems that are closely related to the Tarry–Escott problem. These solutions are also more general and simpler than the solutions of diophantine systems published earlier.

scholarly journals Amount algebras

2021 ◽  
pp. 2250102
Author(s):  
Peyman Nasehpour
Keyword(s):  
Positive Integer ◽  
Valuation Ring ◽  
Prüfer Domain ◽  
Radical Ideal ◽  
Ideal Formula ◽  
Torsion Free ◽  
The Ideal

In this paper, as a generalization to content algebras, we introduce amount algebras. Similar to the Anderson–Badawi [Formula: see text] conjecture, we prove that under some conditions, the formula [Formula: see text] holds for some amount [Formula: see text]-algebras [Formula: see text] and some ideals [Formula: see text] of [Formula: see text], where [Formula: see text] is the smallest positive integer [Formula: see text] that the ideal [Formula: see text] of [Formula: see text] is [Formula: see text]-absorbing. A corollary to the mentioned formula is that if, for example, [Formula: see text] is a Prüfer domain or a torsion-free valuation ring and [Formula: see text] is a radical ideal of [Formula: see text], then [Formula: see text].

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