Topic
Ring of integers
About: Ring of integers is a research topic. Over the lifetime, 1856 publications have been published within this topic receiving 15882 citations.
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TL;DR: The noncommutative Specker Phenomenon was shown to fail if one passes from countable to uncountable in this article, where it was shown that for non-trivial groups G_alpha (alpha in lambda) and uncountably cardinal lambda there are 2π 2π γ 2 γ γ homomorphisms from the complete free product of the G_α's to the ring of integers.
Abstract: Higman proved in 1952 that every free group is non-commutatively slender, this is to say that if G is a free group and h is a homomorphism from the countable complete free product (X_omega Z) to G, then there exists a finite subset F of omega and a homomorphism h:*_{i in F} Z --> G such that h=h rho_F, where rho_F is the natural map from (X_{i in omega})Z to *_{i in F}Z . Corresponding to the abelian case this phenomenon was called the non-commutative Specker Phenomenon. In this paper we show that Higman's result fails if one passes from countable to uncountable. In particular, we show that for non-trivial groups G_alpha (alpha in lambda) and uncountable cardinal lambda there are 2^{2^lambda} homomorphisms from the complete free product of the G_alpha 's to the ring of integers.
4 citations
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TL;DR: In this article, the density of the set of coprime $m$-tuples of algebraic integers is shown to be 1/\zeta_K(m) where k is the Dedekind zeta function of the number field.
Abstract: Let $K$ be a number field with ring of integers $\mathcal O$. After introducing a suitable notion of density for subsets of $\mathcal O$, generalizing that of natural density for subsets of $\mathbb Z$, we show that the density of the set of coprime $m$-tuples of algebraic integers is ${1/\zeta_K(m)}$, where $\zeta_K$ is the Dedekind zeta function of $K$. This generalizes a result by Ces\`aro (1881) concerning the density of coprime pairs in $\mathbb Z$.
4 citations
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15 Dec 2017
TL;DR: The theory of reduced generator matrices to the Hecke rings of matrices over these Euclidean domains are applied and it is shown that if a and b are coprime, then this correspondence is one-to-one.
Abstract: In this study, we consider codes over Euclidean domains modulo their ideals. In the first half of the study, we deal with arbitrary Euclidean domains. We show that the product of generator matrices of codes over the rings mod a and mod b produces generator matrices of all codes over the ring mod a
b
, i.e., this correspondence is onto. Moreover, we show that if a and b are coprime, then this correspondence is one-to-one, i.e., there exist unique codes over the rings mod a and mod b that produce any given code over the ring mod a
b
through the product of their generator matrices. In the second half of the study, we focus on the typical Euclidean domains such as the rational integer ring, one-variable polynomial rings, rings of Gaussian and Eisenstein integers, p-adic integer rings and rings of one-variable formal power series. We define the reduced generator matrices of codes over Euclidean domains modulo their ideals and show their uniqueness. Finally, we apply our theory of reduced generator matrices to the Hecke rings of matrices over these Euclidean domains.
4 citations
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TL;DR: In this article, an arithmetic site of Connes-Consani type for imaginary quadratic number fields with class number 1 is presented. But the main difficulty here is that their constructions and part of their results strongly rely on the natural order existing on real numbers which is compatible with basic arithmetic operations.
4 citations
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TL;DR: In this paper, the authors established the specialization method on characteristic ideals for finitely generated torsion modules over a complete local normal domain R that is module-finite over $O[[x_1,..., x_d]], where O is the ring of integers of a finite extension of the field of p-adic integers $Q_p.
Abstract: The aim of this article is to establish the specialization method on characteristic ideals for finitely generated torsion modules over a complete local normal domain R that is module-finite over $O[[x_1, ..., x_d]]$, where $O$ is the ring of integers of a finite extension of the field of p-adic integers $Q_p$. The specialization method is a technique that recovers the information on the characteristic ideal $char_R(M)$ from $char_{R/I}(M/IM)$, where I varies in a certain family of nonzero principal ideals of R. As applications, we prove Euler system bound over Cohen-Macaulay normal domains by combining the main results in an earlier article of the first named author and then we prove one of divisibilities of the Iwasawa main conjecture for two-variable Hida deformations generalizing the main theorem obtained in an article of the first named author.
4 citations