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In this note, we show that the result [1, Proposition 5.2] is inaccurate. We further give and prove the correct modification of such a result. Some applications are also given.

Let $ (K,\nu) $ be an arbitrary-rank valued field, let $ R_{\nu} $ be the valuation ring of $ (K,\nu) $ , and let $ K(\alpha)/K $ be a separable finite field extension generated over $ K $ by a root of a monic irreducible polynomial $ f\in R_{\nu}[X] $ . We give some necessary and sufficient conditions for $ R_{\nu}[\alpha] $ to be integrally closed. We further characterize the integral closedness of $ R_{\nu}[\alpha] $ which is based on information about the valuations on $ K(\alpha) $ extending $ \nu $ . Our results enhance and generalize some existing results as well as provide applications and examples.

We discuss when an order in a Dedekind domain $$R$$ is equal to $$R$$ (is the maximal order in $$R$$ ). Every order in $$R$$ is a subring of $$R$$ . This fact implies the existence of natural homomorphisms between objects related to orders such that the group of Cartier divisors, the Picard group, the group of Weil divisors, the Chow group and the Witt ring of an order. We examine the maximality of an order in $$R$$ in the context of such natural homomorphisms. In [8], we discuss when an order $$\mathcal{O}$$ in $$R$$ is equal to $$R$$ on the assumption that either the Picard group of $$R$$ or the Picard group of $$\mathcal{O}$$ is a torsion group. In this paper, we abandon this assumption. We formulate equivalent conditions for the maximality of $$\mathcal{O}$$ for any Dedekind domain $$R$$ and any order $$\mathcal{O}$$ in $$R$$ .

In this article we calculate the order projection in a space $$\mathcal{OA}_{r}(E,F)$$ of all regular orthogonally additive operators from a vector lattice $$E$$ to a Dedekind complete vector lattice $$F$$ , onto the band $$\{T\}^{\perp\perp}$$ of $$\mathcal{OA}_{r}(E,F)$$ which is generated by a disjointness preserving orthogonally additive operator $$T\colon E\to F$$ .

In this article we explore orthogonally additive (nonlinear) operators in vector lattices. First we investigate the lateral order on vector lattices and show that with every element $$e$$ of a $$C$$ -complete vector lattice $$E$$ is associated a lateral-to-order continuous orthogonally additive projection $$\mathfrak{p}_{e}\colon E\to\mathcal{F}_{e}$$ . Then we prove that for an order bounded positive $$AM$$ -compact orthogonally additive operator $$S\colon E\to F$$ defined on a $$C$$ -complete vector lattice $$E$$ and taking values in a Dedekind complete vector lattice $$F$$ all elements of the order interval $$[0,S]$$ are $$AM$$ -compact operators as well.

In the present work, we prove few new Dedekind eta-function identities of level 6 discovered by Somos in two different methods. Also during this process, we give an alternate method to Somos’s Dedekind eta-function identities of level 6 proved by B. R. Srivatsa Kumar et. al. As an application of this, we establish colored partition identities.

Let R be a commutative ring with identity and let $\mathcal {X} = \{X_{\lambda }\}_{\lambda \in {\Lambda }}$ be an arbitrary set (either finite or infinite) of indeterminates over R. There are three types of power series rings in the set $\mathcal {X}$ over R, denoted by $R[[\mathcal {X}]]_{i}$ , i = 1,2,3, respectively. In general, $R[[\mathcal X]]_{1} \subseteq R[[\mathcal {X}]]_{2} \subseteq R[[\mathcal {X}]]_{3}$ and the two containments can be strict. For a power series f ∈ R[[X]]3, we denote by Af the ideal of R generated by the coefficients of f. In this paper, we show that a Dedekind–Mertens type formula holds for power series in $R[[\mathcal {X}]]_{3}$ . More precisely, if $g\in R[[\mathcal {X}]]_{3}$ such that the locally minimal number of special generators of Ag is k + 1, then $A_{f}^{k+1}A_{g} = {A_{f}^{k}} A_{fg}$ for all $f \in R[[\mathcal X]]_{3}$ . The same formula holds if f belongs to $R[[\mathcal {X}]]_{i}$ , i = 1,2, respectively. Our result is a generalization of previously known results in which $\mathcal X$ has a single indeterminate or g is a polynomial.

A projection lateral band $G$ in a Riesz space $E$ is defined to be a lateral band which is the image of an orthogonally additive projection $Q: E \to E$ possessing the property that $Q(x)$ is a fragment of $x$ for all $x \in E$, called a lateral retraction of $E$ onto $G$ (which is then proved to be unique). We investigate properties of lateral retracts, that are, images of lateral retractions, and describe lateral retractions onto principal projection lateral bands (i.e. lateral bands generated by single elements) in a Riesz space with the principal projection property. Moreover, we prove that every lateral retract is a lateral band, and every lateral band in a Dedekind complete Riesz space is a projection lateral band.

We study lower bounds of a general family of L-functions on the $1$ -line. More precisely, we show that for any $F(s)$ in this family, there exist arbitrarily large t such that $F(1+it)\geq e^{\gamma _F} (\log _2 t + \log _3 t)^m + O(1)$ , where m is the order of the pole of $F(s)$ at $s=1$ . This is a generalisation of the result of Aistleitner, Munsch and Mahatab [‘Extreme values of the Riemann zeta function on the $1$ -line’, Int. Math. Res. Not. IMRN2019(22) (2019), 6924–6932]. As a consequence, we get lower bounds for large values of Dedekind zeta-functions and Rankin-Selberg L-functions of the type $L(s,f\times f)$ on the $1$ -line.

The goal of this article is to present some method of optimal extension of positive order continuous and $ \sigma $ -order continuous operators on quasi-Banach function spaces with values in Dedekind complete quasi-Banach lattices. The optimal extension of such an operator is the smallest extension of the Bartle–Dunford–Schwartz type integral. It is also shown that if a positive operator sends order convergent sequences to quasinorm convergent sequences, then its optimal extension is the Bartle–Dunford–Schwartz type integral.

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.

In this article, we consider orthogonally additive operators defined on a vector lattice E and taking value in a Banach space X. We say that an orthogonally additive operator $T:E\to X$ is narrow if for every $e\in E$ and $\varepsilon>0$ there exists a decomposition $e=e_1\sqcup e_2$ of e into a sum of two disjoint fragments e1 and e2 such that $\|Te_1-Te_2\

We give a regular extension of the Menon-type identity to residually finite Dedekind domains.

In this paper, we investigate the linkage of ideals, in Noetherian and non-Noetherian settings, with the aim to establish new characterizations of classical notions of domains through linkage theory. Two main results assert that a Noetherian domain is Dedekind if and only if it has the primary linkage property; and a domain is almost Dedekind (resp., Prufer) if and only if it has the linkage (resp., finite linkage) property. Also, we prove that a finite-dimensional valuation domain is a DVR (i.e., Noetherian) if and only if it has the primary linkage property.

For a totally ordered group G we determine the largest group contained in its Dedekind completion G#. It was the result of studying the family of convex subgroups of G and some well-known properties of ordered groups.

We study the relationships between the properties of nonperiodic groups and the norms of their decomposable subgroups. The influence of restrictions imposed on the norms of decomposable subgroups and on the properties of the group is analyzed under the condition that this norm is non-Dedekind and locally nilpotent. We also describe the structure of nonperiodic locally soluble groups for which the norm of decomposable subgroups possesses the indicated properties.

Let D be a commutative domain with identity, and let $${\mathcal {L}}(D)$$ be the lattice of nonzero ideals of D. Say that D is ideal upper finite provided $${\mathcal {L}}(D)$$ is upper finite, that is, every nonzero ideal of D is contained in but finitely many ideals of D. Now let $$\kappa >2^{\aleph _0}$$ be a cardinal. We show that a domain D of cardinality $$\kappa $$ is ideal upper finite if and only if D is a Dedekind domain. We also show (in ZFC) that this result is sharp in the sense that if $$\kappa $$ is a cardinal such that $$\aleph _0\le \kappa \le 2^{\aleph _0}$$ , then there is an ideal upper finite domain of cardinality $$\kappa $$ which is not Dedekind.

The formalization of mathematics based on theorem prover becomes increasingly important in mathematics and computer science, and, particularly, formalizing fundamental mathematical theories becomes especially essential. In this paper, we describe the formalization in Coq of eight very representative completeness theorems of real numbers. These theorems include the Dedekind fundamental theorem, Supremum theorem, Monotone convergence theorem, Nested interval theorem, Finite cover theorem, Accumulation point theorem, Sequential compactness theorem, and Cauchy completeness theorem. We formalize the real number theory strictly following Landau&rsquo, s Foundations of Analysis where the Dedekind fundamental theorem can be proved. We extend this system and complete the related notions and properties for finiteness and sequence. We prove these theorems in turn from Dedekind fundamental theorem, and finally prove the Dedekind fundamental theorem by the Cauchy completeness theorem. The full details of formal proof are checked by the proof assistant Coq, which embodies the characteristics of reliability and interactivity. This work can lay the foundation for many applications, especially in calculus and topology.

One proves that a field carries a compatible directed order if and only if it has characteristic 0 and is real or has a transcendence degree of at least 1 over the field of rational numbers.

Somos conjectured thousands of Dedekind $$\eta $$ -function identities of various levels, around 6200 in number. He did so using computational evidence but has not sought to provide any proof for these identities. In this paper, we prove level 6 of Somos’s Dedekind $$\eta $$ -function identities containing five terms in two methods. Further, as an application of these identities, we deduce colored partitions for the same.

Domain extension in mathematics occurs whenever a given mathematical domain is augmented so as to include new elements. Manders argues that the advantages of important cases of domain extension are captured by the model-theoretic notions of existential closure and model completion. In the specific case of domain extension via ideal elements, I argue, Manders’s proposed explanation does not suffice. I then develop and formalize a different approach to domain extension based on Dedekind’s Habilitationsrede, to which Manders’s account is compared. I conclude with an examination of three possible stances towards extensions via ideal elements.

In this article, ( X , A , μ ) is a measure apace. A classical result establishes a Riesz isomorphism between L 1 ( μ ) ∼ and L ∞ ( μ ) in case the measure μ is σ -finite. In general, there still is a natural Riesz homomorphism Φ : L ∞ ( μ ) → L 1 ( μ ) ∼ , but it may not be injective or surjective. We prove that always the range of Φ is an order dense Riesz subspace of L 1 ( μ ) ∼ . If μ is semi-finite, then L 1 ( μ ) ∼ is a Dedekind completion of L ∞ ( μ ) .

We prove that the category of Dedekind $\sigma$-complete Riesz spaces is an infinitary variety, and we provide an explicit equational axiomatization. In fact, we show that finitely many axioms suffice over the usual equational axiomatization of Riesz spaces. Our main result is that $\mathbb{R}$, regarded as a Dedekind $\sigma$-complete Riesz space, generates this category as a quasi-variety, and therefore as a variety. Analogous results are established for the categories of (i) Dedekind $\sigma$-complete Riesz spaces with a weak order unit, (ii) Dedekind $\sigma$-complete lattice-ordered groups, and (iii) Dedekind $\sigma$-complete lattice-ordered groups with a weak order unit., Comment: 15 pages

In this article, we classify all those indecomposable 2-absorbing multiplication modules with finite-dimensional top over pullback of two Dedekind domains. We extend the definition and results given in [10] to a more general 2-absorbing multiplication modules case.

We show that the second residue map for hermitian Witt groups of an Azumaya algebra A with involution $$\tau $$ of first- or second kind over a semilocal Dedekind domain R is surjective. This proves a generalization to hermitian Witt groups of an exact sequence for Witt groups of quadratic forms due to Springer. If R is a complete discrete valuation ring and $$\tau $$ is of the first kind we show that our short exact sequence of hermitian Witt groups is split. As a corollary we prove a purity theorem for hermitian Witt groups of Azumaya algebras with involutions over a regular semilocal domain of dimension two.

In a Dedekind complete Riesz space, E, we show that if $$(P_n)$$ is a sequence of band projections in E then $$\begin{aligned} \limsup \limits _{n\rightarrow \infty } P_n - \liminf \limits _{n\rightarrow \infty } P_n = \limsup \limits _{n\rightarrow \infty } P_n(I-P_{n+1}). \end{aligned}$$This identity is used to obtain conditional extensions in a Dedekind complete Riesz spaces with weak order unit and conditional expectation operator of the Barndorff-Nielsen and Balakrishnan–Stepanov generalizations of the first Borel–Cantelli theorem.

Let R be a principal ideal domain with quotient field K, and L = K(α), where α is a root of a monic irreducible polynomial F (x) ∈ R[x]. Let ℤ L be the integral closure of R in L. In this paper, for every prime p of R, we give a new efficient version of Dedekind’s criterion in R, i.e., necessary and sufficient conditions on F (x) to have p not dividing the index [ℤ L : R[α]], for every prime p of R. Some computational examples are given for R = ℤ.

We investigate some properties of strongly order bounded operators. For example, we prove that if a Riesz space E is an ideal in $$E^{\sim \sim }$$ and F is a Dedekind complete Riesz space then for each ideal A of E, T is strongly order bounded on A if and only if $$T_A$$ is strongly order bounded. We show that the class of strongly order bounded operators satisfies the domination problem. On the other hand, we present two ways for decomposition of strongly order bounded operators, and we give some of their properties. Also, it is shown that E has order continuous norm or F has the b-property whenever each pre-regular operator form E into F is order bounded.

Lattice is a partially ordered set in which all finite subsets have a least upper bound and greatest lower bound. Dedekind worked on lattice theory in the 19th century. Nano topology explored by Lellis Thivagar et.al. can be described as a collection of nano approximations, a non-empty finite universe and empty set for which equivalence classes are buliding blocks. This is named as Nano topology, because of its size and what ever may be the size of universe it has atmost five elements in it. The elements of Nano topology are called the Nano open sets. This paper is to study the nano topology within the context of lattices. In lattice, there is a special class of joincongruence relation which is defined with respect to an ideal. We have defined the nano approximations of a set with respect to an ideal of a lattice. Also some properties of the approximations of a set in a lattice with respect to ideals are studied. On the other hand, the lower and upper approximations have also been studied within the context various algebraic structures.

We prove, in particular, that if E is a Dedekind complete atomless Riesz space and X is a Banach space then the sum of a narrow and a C-compact laterally continuous orthogonally additive operators from E to X is narrow. This generalizes in several directions known results on narrowness of the sum of a narrow and a compact operators for the settings of linear and orthogonally additive operators defined on Kothe function spaces and Riesz spaces.

Let $${\widetilde{k}}$$ be a fixed cubic field, F a quadratic field and $$L=\widetilde{k}\cdot F$$. In this paper and its companion paper, we determine the density of more or less the ratio of the residues of the Dedekind zeta functions of L, F where F runs through quadratic fields.

Let $T=\left (\begin {array}{cc} R & V\\ W & S \end {array}\right )$ be the ring of a Morita context such that VW = R and WV = S. In this paper, we give necessary and sufficient conditions for T to be a G-Dedekind prime 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.

We study $b$-property of a sublattice (or an order ideal) $F$ of a vector lattice $E$. In particular, $b$-property of $E$ in $E^\delta$, the Dedekind completion of $E$, $b$-property of $E$ in $E^u$, the universal completion of $E$, and $b$-property of $E$ in $\hat{E}(\hat{\tau})$, the completion of $E$.

For a Dedekind domain $D$, let $\mathcal{P}(D)$ be the set of ideals of $D$ that are radical of a principal ideal. We show that, if $D,D'$ are Dedekind domains and there is an order isomorphism between $\mathcal{P}(D)$ and $\mathcal{P}(D')$, then the rank of the class groups of $D$ and $D'$ is the same.

The groups having exactly one normalizer are Dedekind groups. All finite groups with exactly two normalizers were classified by Perez-Ramos in 1988. In this paper we prove that every finite group with at most 26 normalizers of $$\{2,3,5\}$$ -subgroups is soluble and we also show that every finite group with at most 21 normalizers of cyclic $$\{2,3,5\}$$ -subgroups is soluble. These confirm Conjecture 3.7 of Zarrin (Bull Aust Math Soc 86:416–423, 2012).

We study d-CS-Rickart modules (i.e. modules M such that for every endomorphism φ of M, the image of φ lies above a direct summand of M) over Dedekind domains. The structure of d-CS-Rickart modules over discrete valuation rings is fully determined. It is also shown that for a d-CS-Rickart R-module M over a nonlocal Dedekind domain R, the following assertions hold

We consider random polynomials with independent identically distributed coefficients with a fixed law. Assuming the Riemann hypothesis for Dedekind zeta functions, we prove that such polynomials are irreducible and their Galois groups contain the alternating group with high probability as the degree goes to infinity. This settles a conjecture of Odlyzko and Poonen conditionally on RH for Dedekind zeta functions., 50 pages, this is the accepted version for publication in Acta Math., minor changes and corrections based on referees' reports

We study the number of non-trivial simple zeros of the Dedekind zeta-function of a quadratic number field in the rectangle established by Conrey et al [Simple zeros of the zeta function of a quadratic number field. I. Invent. Math.86 (1986), 563–576].

Let $\mathfrak{D}$ be a residually finite Dedekind domain and let $\mathfrak{n}$ be a nonzero ideal of $\mathfrak{D}$. We consider counting problems for the ideal chains in $\mathfrak{D}/\mathfrak{n}$. By using the Cauchy–Frobenius–Burnside lemma, we also obtain some further extensions of Menon’s identity.

In this paper, we describe properties of the Dedekind η-function, constructions arising from it, and their applications in various topics of the number theory and algebra. We discuss connections with the group representation theory and the study of the structure of spaces of modular forms. Special attention is paid to the special class of modular forms, namely, η-functions with multiplicative coefficients.

The squared absolute value of the Dedekind zeta functions of imaginary quadratic fields of class number one on the critical line is expressed in terms of averaged values associated with the spectrum of the automorphic Laplacian with respect to the full modular group.

In this paper, we characterize the secondary modules and classify the indecomposable secondary modules over Dedekind domains. We also prove that every secondary module over a Dedekind domain is pure injective.

The main purpose of this paper is to define p-adic and q-Dedekind type sums. Using the Volkenborn integral and the Teichmüller character representations of the Bernoulli polynomials, we give reciprocity law of these sums. These sums and their reciprocity law generalized some of the classical p-adic Dedekind sums and their reciprocity law. It is to be noted that the Dedekind reciprocity laws, is a fine study of the existing symmetry relations between the finite sums, considered in our study, and their symmetries through permutations of initial parameters.