Dedekind Ring Research Articles

On ϕ-(weak) global dimension

In this paper, we will introduce and study the homological dimensions defined in the context of commutative rings with prime nilradical. So all rings considered in this paper are commutative with identity and with prime nilradical. We will introduce a new class of modules which are called [Formula: see text]-u-projective which generalizes the projectivity in the classical case and which is different from those introduced by the authors of [Y. Pu, M. Wang and W. Zhao, On nonnil-commutative diagrams and nonnil-projective modules, Commun. Algebra, doi:10.1080/00927872.2021.2021223; W. Zhao, On [Formula: see text]-exact sequence and [Formula: see text]-projective module, J. Korean Math. 58(6) (2021) 1513–1528]. Using the notion of [Formula: see text]-flatness introduced and studied by the authors of [G. H. Tang, F. G. Wang and W. Zhao, On [Formula: see text]-Von Neumann regular rings, J. Korean Math. Soc. 50(1) (2013) 219–229] and the nonnil-injectivity studied by the authors of [W. Qi and X. L. Zhang, Some Remarks on [Formula: see text]-Dedekind rings and [Formula: see text]-Prüfer rings, preprint (2022), arXiv:2103.08278v2 [math.AC]; X. Y. Yang, Generalized Noetherian Property of Rings and Modules (Northwest Normal University Library, Lanzhou, 2006); X. L. Zhang, Strongly [Formula: see text]-flat modules, strongly nonnil-injective modules and their homological dimensions, preprint (2022), https://arxiv.org/abs/2211.14681; X. L. Zhang and W. Zhao, On Nonnil-injective modules, J. Sichuan Normal Univ. 42(6) (2009) 808–815; W. Zhao, Homological theory over NP-rings and its applications (Sichuan Normal University, Chengdu, 2013)], we will introduce the [Formula: see text]-injective dimension, [Formula: see text]-projective dimension and [Formula: see text]-flat dimension for modules, and also the [Formula: see text]-(weak) global dimension of rings. Then, using these dimensions, we characterize several [Formula: see text]-rings ([Formula: see text]-Prüfer, [Formula: see text]-chained, [Formula: see text]-von Neumann, etc). Finally, we study the [Formula: see text]-(weak) global dimension of the trivial ring extensions defined by some conditions.

A Bloch–Ogus theorem for henselian local rings in mixed characteristic

We show a conditional exactness statement for the Nisnevich Gersten complex associated to an mathbb {A}^1-invariant cohomology theory with Nisnevich descent for smooth schemes over a Dedekind ring with only infinite residue fields. As an application we derive a Nisnevich analogue of the Bloch–Ogus theorem for étale cohomology over a henselian discrete valuation ring with infinite residue field.

On the Bhargava factorial of polynomial maps

Bhargava introduced a generalization of the factorial function to extend classical results in integers to Dedekind rings and unify them. We study the Bhargava factorial of the images of polynomial maps from an analytic perspective. We first give the -adic closures of the images of polynomial maps, which is the key to compute -adic part of the Bhargava factorial. Then, as a special case, we give the Stirling formula for the image of quadratic polynomials with integer coefficients.

On K0 of locally finite categories

We calculate the Grothendieck group K0(A), where A is an additive category, locally finite over a Dedekind ring and satisfying some additional conditions. The main examples are categories of modules over finite algebras and the stable homotopy category SW of finite CW-complexes. We show that this group is a direct sum of a free group arising from localizations of the category A and a group analogous to the groups of ideal classes of maximal orders. As a corollary, we obtain a new simple proof of the Freyd's theorem describing the group K0(SW).

A Polynomial Composites and Monoid Domains as Algebraic Structures and their Applications

This paper contains the results collected so far on polynomial composites in terms of many basic algebraic properties. Since it is a polynomial structure, results for monoid domains come in here and there. The second part of the paper contains the results of the relationship between the theory of polynomial composites, the Galois theory and the theory of nilpotents. The third part of this paper shows us some crypto systems. We find generalizations of known ciphers taking into account the infinite alphabet and using simple algebraic methods. We also find two cryptosystems in which the structure of Dedekind rings resides, namely certain elements are equivalent to fractional ideals. Finally, we find the use of polynomial composites and monoid domains in cryptology.

THE GROTHENDIECK–SERRE CONJECTURE OVER SEMILOCAL DEDEKIND RINGS

For a reductive group scheme G over a semilocal Dedekind ring R with total ring of fractions K, we prove that no nontrivial G-torsor trivializes over K. This generalizes a result of Nisnevich–Tits, who settled the case when R is local. Their result, in turn, is a special case of a conjecture of Grothendieck–Serre that predicts the same over any regular local ring. With a patching technique and weak approximation in the style of Harder, we reduce to the case when R is a complete discrete valuation ring. Afterwards, we consider Levi subgroups to reduce to the case when G is semisimple and anisotropic, in which case we take advantage of Bruhat–Tits theory to conclude. Finally, we show that the Grothendieck–Serre conjecture implies that any reductive group over the total ring of fractions of a regular semilocal ring S has at most one reductive S-model.

Commutators of Congruence Subgroups in the Arithmetic Case

In a joint paper of the author with Alexei Stepanov, it was established that for any two comaximal ideals A and B of a commutative ring R, A + B = R, and any n ≥ 3 one has [E(n,R,A),E(n,R,B)] = E(n,R,AB). Alec Mason and Wilson Stothers constructed counterexamples demonstrating that the above equality may fail when A and B are not comaximal, even for such nice rings as ℤ [i]. The present note proves a rather striking result that the above equality and, consequently, also the stronger equality [GL(n,R,A), GL(n,R,B)] = E(n,R,AB) hold whenever R is a Dedekind ring of arithmetic type with infinite multiplicative group. The proof is based on elementary calculations in the spirit of the previous papers by Wilberd van der Kallen, Roozbeh Hazrat, Zuhong Zhang, Alexei Stepanov, and the author, and also on an explicit computation of the multirelative SK1 from the author’s paper of 1982, which, in its turn, relied on very deep arithmetical results by Jean-Pierre Serre and Leonid Vaserstein (as corrected by Armin Leutbecher and Bernhard Liehl). Bibliography: 50 titles.

On Syzygy Modules over Laurent Polynomial Rings

In this paper, we present a dynamical method for computing the syzygy module of multivariate Laurent polynomials with coefficients in a Dedekind ring (with zero divisors) by reducing the computation over Laurent polynomial rings to calculations over a polynomial ring via an appropriate isomorphism.

Analysis of the RSA-cryptosystem in abstract number rings

Quantum computers can be a real threat to some modern cryptosystems (such as the RSA-cryptosystem). The analogue of the RSA-cryptosystem in abstract number rings is not affected by this threat, as there are currently no factorization algorithms using quantum computing for ideals. In this paper considered an analogue of RSA-cryptosystem in abstract number rings. Proved the analogues of theorems related to its cryptographic strength. In particular, an analogue of Wiener’s theorem on the small secret exponent is proved. The analogue of the re-encryption method is studied. On its basis the necessary restrictions on the parameters of the cryptosystem are obtained. It is also shown that in numerical Dedekind rings the factorization problem is polynomial equivalent to factorization in integers.

Polynomial index in discrete valuation rings

Abstract Let R be a discrete valuation ring, its nonzero prime ideal, P ∈R[X] a monic irreducible polynomial, and K the quotient field of R. We give in this paper a lower bound for the -adic valuation of the index of P over R in terms of the degrees of the monic irreducible factors of the reduction of P modulo . By localization, the same result holds true over Dedekind rings. As an important immediate application, when the lower bound is greater than zero, we conclude that no root of P generates a power basis for the integral closure of R in the field extension of K defined by P.

Strongly graded rings which are generalized Dedekind rings

Let [Formula: see text] be a strongly graded ring of type [Formula: see text] such that [Formula: see text] is a prime Goldie ring with its quotient ring [Formula: see text]. It is shown that the following three conditions are equivalent: (i) [Formula: see text] is a [Formula: see text]-invariant generalized Dedekind ring ([Formula: see text]-Dedekind ring for short), (ii) [Formula: see text] is a [Formula: see text]-Dedekind ring and (iii) [Formula: see text] is a graded [Formula: see text]-Dedekind ring. We describe all invertible ideals of [Formula: see text]-Dedekind rings in terms of [Formula: see text] and [Formula: see text]. We provide counterexamples of [Formula: see text]-invariant [Formula: see text]-Dedekind rings which are not [Formula: see text]-Dedekind rings.

First Nontrivial Group of Homologies for the Simplicial Schemes of Unimodular Frames Over the Dedekind Ring

We prove a theorem on the generation of the first nontrivial group of homologies of a simplicial scheme of unimodular frames over the Dedekind ring by standard cycles.

T-structures on hereditary categories

We study aisles, equivalently t-structures, in the derived category of a hereditary abelian category. Given an aisle, we associate a sequence of subcategories of the abelian category by considering the different homologies of the aisle. We then obtain a sequence, called a narrow sequence. We then prove that a narrow sequence in a hereditary abelian category consists of a nondecreasing sequence of wide subcategories, together with a tilting torsion class in each of these wide subcategories. Studying the extra conditions that the narrow sequences coming from aisles must satisfy we get a bijection between coreflective narrow seqeunces and t-structures in the derived category. In some cases, including the case of finite-dimensional modules over a finite-dimensional hereditary algebra, we refine our results and show that the t-structures are determined by an increasing sequence of coreflective wide subcategories together with a tilting torsion class in the orthogonal of one wide subcategory in the next, effectively decoupling the different tilting torsion theories one chooses in the wide subcategories. These results are sufficient to recover known classifications of t-structures for smooth projective curves, and for finitely generated modules over a Dedekind ring.

BL-rings

The main goal of this article is to introduce BL-rings, i.e., commutative rings whose lattices of ideals can be equipped with a structure of BL-algebra. We obtain a description of such rings, and study the connections between the new class and well known classes such as multiplications rings, Baer rings, Dedekind rings.

ОДНОРОДНЫЕ ОТОБРАЖЕНИЯ СМЕШАННЫХ МОДУЛЕЙ

In this paper we study mixed modules, with the following property: every homogeneous function of several variables of a module is additive. By a homogeneous function we mean any mapping of the direct sum of a finite number of copies of a module into the module itself that commutes with the endomorphisms of the given module. In the universal algebra, the algebraic structure is said to be endoprimal if all its term-functions commute with endomorphisms. It is well-known that each endodualizable finite algebra is endoprimal. Some authors have studied endoprimal algebras in varieties of vector spaces, semilattices, Boolean algebras, Stone algebras, Heyting algebras, and Abelian groups. In this article, the links between endoprimality and the properties of the multiplicative semigroup of the endomorphism ring of a module, which the author started earlier. Classes of mixed non-reduced splitting modules and reduced modules over commutative Dedekind ring have been investigated. Links between this problem and the property of unique additivity has been shown.

Tannakian duality over Dedekind rings and applications

We establish a duality between flat affine group schemes and rigid tensor categories equipped with a neutral fiber functor (called Tannakian lattice), both defined over a Dedekind ring. We use this duality and the known Tannakian duality due to Saavedra to study morphisms between flat affine group schemes. Next, we apply our new duality to the category of stratified sheaves on a smooth scheme over a Dedekind ring R to define the relative differential fundamental group scheme of the given scheme and compare the fibers of this group scheme with the fundamental group scheme of the fibers. When R is a complete DVR of equal characteristic we show that this category is Tannakian in the sense of Saavedra.

On the flatness and the projectivity over Hopf subalgebras of Hopf algebras over Dedekind rings

We study the flatness and the projectivity of Hopf algebras, defined over a Dedekind ring, over their Hopf subalgebras. We give a criterion for the faithful flatness and use it to show the faithful flatness of an arbitrary flat Hopf algebra upon its finite normal Hopf subalgebras. For the projectivity over Hopf subalgebras of a Hopf algebra we need some finiteness condition in terms of the module of integrals. In particular we show that the module of integrals is projective of rank one.

RSA cryptosystem for dedekind rings

May soon be invented quantum computer and some known cryptosystems (such as RSA) will be threatened breaking. This paper is aimed at establishing necessary conditions for the maximum possible algebraic generalization of the classical RSA algorithm. We substitute ideals of a Dedekind ring for integers. Ideals in Dedekind rings allow the unique decomposition into a product of maximal ideals, but may not be the principal ideals. Also we define Euler’s ϕ-function for ideal of a Dedekind ring and describe some properties of this function. We hope that our proposed method will help to develop algorithms for encryption, which is hard to crack using a quantum computer.

ON 𝜙-SHARP RINGS

The purpose of this paper is to introduce some new class of rings that are closely related to the classes of sharp domains, pseudo-Dededkind domains, TV domains and finite character domains. A ring R is called a <TEX>${\phi}$</TEX>-sharp ring if whenever for nonnil ideals I, A, B of R with <TEX>$I{\supseteq}AB$</TEX>, then I = A'B' for nonnil ideals A', B' of R where <TEX>$A^{\prime}{\supseteq}A$</TEX> and <TEX>$B^{\prime}{\supseteq}B$</TEX>. We proof that a <TEX>${\phi}$</TEX>-Dedekind ring is a <TEX>${\phi}$</TEX>-sharp ring and we get some properties that by them a <TEX>${\phi}$</TEX>-sharp ring is a <TEX>${\phi}$</TEX>-Dedekind ring.

On Strong Indecomposability of the Dedekind Ring Localization

Abstract. In this work, we study the question of sufficient conditions for strong indecomposability of Dedekind rings as finite rank modules over their Dedekind subrings. This question was considered earlier in the special case of rings of algebraic integers. Necessary and sufficient conditions are given for Galois extensions. The question of k-indecomposability is also considered.