共查询到20条相似文献,搜索用时 31 毫秒
1.
Alexandra Shlapentokh 《Transactions of the American Mathematical Society》2008,360(7):3541-3555
Using Poonen's version of the ``weak vertical method' we produce new examples of ``large' and ``small' rings of algebraic numbers (including rings of integers) where and/or the ring of integers of a subfield are existentially definable and/or where the ring version of Mazur's conjecture on the topology of rational points does not hold.
2.
Thomas Scanlon 《Journal of the American Mathematical Society》2008,21(3):893-908
Using the work of several other mathematicians, principally the results of Poonen refining the work of Pop that algebraic independence is definable within the class of finitely generated fields and of Rumely that the ring of rational integers is uniformly interpreted in global fields, and a theorem on the definability of valuations on function fields of curves, we show that each infinite finitely generated field considered in the ring language is parametrically biinterpretable with . As a consequence, for any finitely generated field there is a first-order sentence in the language of rings which is true in that field but false in every other finitely generated field and, hence, Pop's conjecture that elementarily equivalent finitely generated fields are isomorphic is true.
3.
For a ring extension ${R \subset S, \,(R, S)}$ is called a principal ideal domain pair (for short PID pair) if every domain ${T, \,R \subseteq T \subseteq S}$ , is a principal ideal domain. When R is a field it is shown that (R, S) is a PID pair iff S is algebraic over R. When R is not a field it is proved that the only PID pairs are those such that R is a PID and S is an overring of R. The second purpose of this paper is to study maximal non-PID subrings. We characterize these type of rings. Further applications and results are also presented. 相似文献
4.
K. Varadarajan 《Proceedings Mathematical Sciences》2000,110(2):133-136
Define a ringA to be RRF (respectively LRF) if every right (respectively left)A-module is residually finite. We determine the necessary and sufficient conditions for a formal triangular matrix ring
to be RRF (respectively LRF). Using this we give examples of RRF rings which are not LRF. 相似文献
5.
6.
Let K be a field and
a non-trivial valuation ring of K withm as its maximal ideal. Denote by
and
the rings of polynomials f∈K[X] and rational functions f∈K(X) resp. such that
. We prove that for one variable X we have
if and only if the completion of (K,
) is locally compact or algebraically closed. In the second case—i.e. if K is dense in the algebraic closure of (K,
)—we even get
for any number of variables X=(X1,...,Xn).
This work contains parts of the second author's thesis [Ri] written under the supervision of the first author. 相似文献
7.
Yao Chen ZHU 《数学学报(英文版)》2007,23(1):17-22
In this paper the generalized Mahler type number Mh(g;A,T) is defined, and in the case of multiplicatively dependent parameters gi, hi(1 ≤ i ≤ s) the algebraic independence of the numbers Mhi (gi; A, T)(1 ≤ i ≤ s) is proved, where A and T are certain infinite sequences of non-negative integers and of positive integers, respectively. Furthermore, the algebraic independence result on values of a certain function connected with the generalized Mahler type number and its derivatives at algebraic numbers is also given. 相似文献
8.
Motivated by the categorical notion of localizations applied to the quasi-category of abelian groups, we call a homomorphism
α: A → B a quasi-localization of abelian groups if for each ϕ ∈ Hom(A,B) there is an n ∈ ℕ and a unique ψ ∈ End(B) such that nϕ = ψ ∘ α. In this case we call B a quasi-localization of A. In this paper we investigate quasi-localizations of the integers ℤ. While it is well-known that localizations of ℤ are just
the E-rings, quasi-localizations of ℤ are much more abundant; an injection α: ℤ → M with M torsion-free, is a quasi-localization if and only if, for R = End(M), one has
. We call R the ring of the quasi-localization M. Some old results due to Zassenhaus and Butler show that all rings with free additive groups of finite rank are indeed rings
of quasi-localizations of ℤ. We will extend this result and show that there are also rings of infinite rank with this property.
While there are many realization results of rings R as endomorphism rings of torsion-free abelian groups M in the literature, the group M is usually not contained in the divisible hull of R
+, as is required here. We will use a particular case of a category of left R-modules M with a distinguished family
of submodules and thus
. We will restrict our discussion to the case M = R such that
, and in this case we call the family
of left ideals E-forcing, not to be confused with the notion of forcing in set theory. We will provide many examples of quasi-localizations
M of ℤ, among them those of infinite rank as well as matrix rings for various rings of finite rank. 相似文献
9.
Nigel Byott 《manuscripta mathematica》1991,73(1):289-311
LetL/K be a totally ramified, finite abelian extension of local fields, let
and
be the valuation rings, and letG be the Galois group. We consider the powers
of the maximal ideal of
as modules over the group ring
. We show that, ifG has orderp
m
(withp the residue field characteristic), ifG is not cyclic (or ifG has orderp), and if a certain mild hypothesis on the ramification ofL/K holds, then
and
are isomorphic iffr≡r′ modp
m
. We also give a generalisation of this result to certain extensions not ofp-power degree, and show that, in the casep=2, the hypotheses thatG is abelian and not cyclic can be removed. 相似文献
10.
Steven T. Dougherty T. Aaron Gulliver Masaaki Harada 《Journal of Algebraic Combinatorics》1999,9(3):233-250
In this paper, we investigate self-dual codes over finite rings, specifically the ring
of integers modulo 2m. Type II codes over
are introduced as self-dual codes with Euclidean weights which are a multiple of 2m +1. We describe a relationship between Type II codes and even unimodular lattices. This relationship provides much information on Type II codes. Double circulant Type II codes over
are also studied. 相似文献
11.
Nicolae Popescu Marian Vâjâitu Alexandru Zaharescu 《Algebras and Representation Theory》2006,9(1):47-66
Let T be a transcendental element of
and
the orbit of T. On
we have a Haar measure
. The goal of this paper is to characterize all the elements of
for which the integral
, called the trace of T, is well defined.Presented by A. Verschoren 相似文献
12.
Kamran Divaani-Aazar 《Proceedings Mathematical Sciences》2009,119(1):23-35
Let R be a (not necessarily local) Noetherian ring and M a finitely generated R-module of finite dimension d. Let be an ideal of R and denote the intersection of all prime ideals . It is shown that
where for an Artinian R-module A we put
A. As a consequence, it is proved that for all ideals of R, there are only finitely many non-isomorphic top local cohomology modules having the same support. In addition, we establish an analogue of the Lichtenbaum-Hartshorne vanishing theorem over rings
that need not be local. 相似文献
13.
David J. Grynkiewicz 《Combinatorica》2006,26(4):445-453
An n-set partition of a sequence S is a collection of n nonempty subsequences of S, pairwise disjoint as sequences, such that every term of S belongs to exactly one of the subsequences, and the terms in each subsequence are all distinct so that they can be considered
as sets. If S is a sequence of m+n−1 elements from a finite abelian group G of order m and exponent k, and if
is a sequence of integers whose sum is zero modulo k, then there exists a rearranged subsequence
of S such that
. This extends the Erdős–Ginzburg–Ziv Theorem, which is the case when m = n and wi = 1 for all i, and confirms a conjecture of Y. Caro. Furthermore, we in part verify a related conjecture of Y. Hamidoune, by showing that
if S has an n-set partition A=A1, . . .,An such that |wiAi| = |Ai| for all i, then there exists a nontrivial subgroup H of G and an n-set partition A′ =A′1, . . .,A′n of S such that
and
for all i, where wiAi={wiai |ai∈Ai}. 相似文献
14.
Michel Weber 《Results in Mathematics》2007,51(1-2):141-179
Let where are independent Bernoulli random variables. In relation with the divisor problem, we evaluate the almost sure asymptotic
order of the sums , where and is a sequence of positive integers.
Received: May 23, 2007. Revised: June 8, 2007. 相似文献
15.
Stefan Gille 《Archiv der Mathematik》2007,88(4):333-343
Let
be a closed subscheme of the noetherian scheme X. We show that if X has a dualizing complex
then there exists a dualizing complex
of Z such that there is an isomorphism of coherent Witt groups
for all
.
Received: 3 March 2006 相似文献
16.
A note on nil power serieswise Armendariz rings 总被引:1,自引:0,他引:1
Sana Hizem 《Rendiconti del Circolo Matematico di Palermo》2010,59(1):87-99
A ring R is called nil power serieswise Armendariz if $
\forall f = \sum\limits_{i = 0}^\infty {a_i X^i }
$
\forall f = \sum\limits_{i = 0}^\infty {a_i X^i }
and $
g = \sum\limits_{i = 0}^\infty {b_i X^i }
$
g = \sum\limits_{i = 0}^\infty {b_i X^i }
in R[[X]] such that f g ∈ Nil(R)[[X]], then a
i
b
j
∈ Nil(R) for all i and j. In this note we characterize completely nil power serieswise Armendariz rings with their nilradical Nil(R) (where the nilradical is the set of nilpotent elements). We prove that a ring is nil power serieswise Armendariz if and
only if Nil(R) is an ideal of R. We prove that each power serieswise Armendariz ring is nil power serieswise Armendariz and we give examples of nil power
serieswise Armendariz rings. 相似文献
17.
I. A. Popovyan 《Mathematical Notes》2006,80(1-2):72-82
In the present paper, a polynomial algorithm is suggested for reducing the problem of taking the discrete logarithm in the ring of algebraic integers modulo a power of a prime ideal to a similar problem with the power equal to one. Explicit formulas are obtained; instead of the Fermat quotients, in the case of residues in the ring of rational integers, these formulas use other polynomially computable logarithmic functions, like the $\mathfrak{p}$ -adic logarithm. 相似文献
18.
A. N. Kochubei 《Ukrainian Mathematical Journal》2005,57(5):794-805
In earlier papers, the author studied some classes of equations with Carlitz derivatives for
-linear functions, which are the natural function-field counterparts of linear ordinary differential equations. Here we consider
equations containing self-compositions u ∘ u ... ∘ u of an unknown function. As an algebraic background, imbeddings of the composition ring of
-linear holomorphic functions into skew fields are considered.
__________
Published in Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 5, pp. 669–678, May, 2005. 相似文献
19.
In this work, we investigate linear codes over the ring ${\mathbb{F}_2+u\mathbb{F}_2+v\mathbb{F}_2+uv\mathbb{F}_2}$ . We first analyze the structure of the ring and then define linear codes over this ring which turns out to be a ring that is not finite chain or principal ideal contrary to the rings that have hitherto been studied in coding theory. Lee weights and Gray maps for these codes are defined by extending on those introduced in works such as Betsumiya et al. (Discret Math 275:43–65, 2004) and Dougherty et al. (IEEE Trans Inf 45:32–45, 1999). We then characterize the ${\mathbb{F}_2+u\mathbb{F}_2+v\mathbb{F}_2+uv\mathbb{F}_2}$ -linearity of binary codes under the Gray map and give a main class of binary codes as an example of ${\mathbb{F}_2+u\mathbb{F}_2+v\mathbb{F}_2+uv\mathbb{F}_2}$ -linear codes. The duals and the complete weight enumerators for ${\mathbb{F}_2+u\mathbb{F}_2+v\mathbb{F}_2+uv\mathbb{F}_2}$ -linear codes are also defined after which MacWilliams-like identities for complete and Lee weight enumerators as well as for the ideal decompositions of linear codes over ${\mathbb{F}_2+u\mathbb{F}_2+v\mathbb{F}_2+uv\mathbb{F}_2}$ are obtained. 相似文献
20.
Let A be a Noetherian local ring and I an ideal of A. In this paper we use a slightly generalized notion of a symbolic power
I(n) of I and considerR =
. First we characterize the property of R to be Noetherian by an equimultiplicity condition of some symbolic power I(k). The main purpose of this note is to explore the problem whenR,R′ =
and
are Cohen-Macaulay or Gorenstein algebras in the case that A is a normal domain and ht I=1.
Partially supported by the Max-Planck-Institute of Mathematics Bonn.
Supported by a grant of the Heinrich Hertz-Stiftung.
Supported by a grant of the Alexander von Humboldt-Stiftung. 相似文献