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It is a difficult question to generalize Gauss sums to a ring of algebraic integers of an arbitrary algebraic number field. In this paper, we define and discuss Gauss sums over a Dedekind domain of finite norm. In particular, we give a Davenport–Hasse type formula for some special Gauss sums. As an application, we give some more precise formulas for Gauss sums over the algebraic integer ring of an algebraic number field (see Theorems 4.1 and 4.2).

Heim and Neuhauser investigated some polynomials related to the Dedekind function. They proved the log-concavity of these polynomials and conjectured that they have only real zeros. We prove this conjecture, and deduce some identities for Fibonacci numbers and determine the modes of another sequence related to Fibonacci polynomials.

Around 1991, J. M. Borwein and P. B. Borwein introduced three cubic theta functions a(q), b(q) and c(q) and discovered many interesting identities associated with these functions. The cubic theta functions b(q) and c(q) have product representations and these representations were first established using the theory of modular forms. The first elementary proof of the product representation of b(q) was discovered in 1994 by the Borweins and F. G. Garvan using one of Euler’s identity. They then derived the product representation of c(q) using transformation formulas of Dedekind’s $$\eta (\tau )$$ and some elementary identities satisfied by a(q), b(q) and c(q). In this note, we present three proofs of the product representation of c(q) without the use of the transformation of Dedekind’s $$\eta $$ -function. We also discuss the connections between these proofs and the works of Baruah and Nath (Proc Am Math Soc 142:441–448, 2014) and Ye (Int J Number Theory 12(7):1791–1800, 2016). We also adopt the idea of the Borweins and Garvan to derive the product representation of Jacobi theta function $$\vartheta _4(0|\tau )$$ which leads to a proof of the Jacobi triple product identity.

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.

Fisher [Fis75] and Baur [Bau75] showed independently in the seventies that if $T$ is a complete first-order theory extending the theory of modules, then the class of models of $T$ with pure embeddings is stable. In [Maz4, 2.12], it is asked if the same is true for any abstract elementary class $(K, \leq_p)$ such that $K$ is a class of modules and $\leq_p$ is the pure submodule relation. In this paper we give some instances where this is true: $\textbf{Theorem.}$ Assume $R$ is an associative ring with unity. Let $(K, \leq_p)$ be an AEC such that $K \subseteq R\text{-Mod}$ and $K$ is closed under finite direct sums, then: - If $K$ is closed under pure-injective envelopes, then $(K, \leq_p)$ is $\lambda$-stable for every $\lambda \geq LS(K)$ such that $\lambda^{|R| + \aleph_0}= \lambda$. - If $K$ is closed under pure submodules and pure epimorphic images, then $(K, \leq_p)$ is $\lambda$-stable for every $\lambda$ such that $\lambda^{|R| + \aleph_0}= \lambda$. - Assume $R$ is Von Neumann regular. If $K$ is closed under submodules and has arbitrarily large models, then $(K, \leq_p)$ is $\lambda$-stable for every $\lambda$ such that $\lambda^{|R| + \aleph_0}= \lambda$. As an application of these results we give new characterizations of noetherian rings, pure-semisimple rings, dedekind domains, and fields via superstability. Moreover, we show how these results can be used to show a link between being good in the stability hierarchy and being good in the axiomatizability hierarchy. Another application is the existence of universal models with respect to pure embeddings in several classes of modules. Among them, the class of flat modules and the class of injective torsion modules., Comment: 22 pages

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.

In this investigation we give a module-theoretic counterpart of the well known De Morgan’s laws for rings and topological spaces. We observe that the module-theoretic De Morgan’s laws are related with semiprime modules and modules in which the annihilator of any fully invariant submodule is a direct summand. Also, we give a general treatment of De Morgan’s laws for ordered structures (idiomatic-quantales). At the end, the manuscript goes back to the ring theoretic realm, in this case we study the non-commutative counterpart of Dedekind domains, and we describe Asano prime rings using the strong De Morgan law.

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 $$R$$ .

On a widespread naturalist view, the meanings of mathematical terms are determined, and can only be determined, by the way we use mathematical language—in particular, by the basic mathematical principles we’re disposed to accept. But it’s mysterious how this can be so, since, as is well known, minimally strong first-order theories are non-categorical and so are compatible with countless non-isomorphic interpretations. As for second-order theories: though they typically enjoy categoricity results—for instance, Dedekind’s categoricity theorem for second-order and Zermelo’s quasi-categoricity theorem for second-order —these results require full second-order logic. So appealing to these results seems only to push the problem back, since the principles of second-order logic are themselves non-categorical: those principles are compatible with restricted interpretations of the second-order quantifiers on which Dedekind’s and Zermelo’s results are no longer available. In this paper, we provide a naturalist-friendly, non-revisionary solution to an analogous but seemingly more basic problem—Carnap’s Categoricity Problem for propositional and first-order logic—and show that our solution generalizes, giving us full second-order logic and thereby securing the categoricity or quasi-categoricity of second-order mathematical theories. Briefly, the first-order quantifiers have their intended interpretation, we claim, because we’re disposed to follow the quantifier rules in an open-ended way. As we show, given this open-endedness, the interpretation of the quantifiers must be permutation-invariant and so, by a theorem recently proved by Bonnay and Westerståhl, must be the standard interpretation. Analogously for the second-order case: we prove, by generalizing Bonnay and Westerståhl’s theorem, that the permutation invariance of the interpretation of the second-order quantifiers, guaranteed once again by the open-endedness of our inferential dispositions, suffices to yield full second-order logic.

Michael Somos used PARI/GP script to generate several Dedekind eta-function identities by using computer. In the present work, we prove two new Dedekind eta-function identities of level 10 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 10 proved by B. R. Srivatsa Kumar and D. Anu Radha. As an application of this, we establish colored partition identities.

For decades critics have categorised John Gay's Trivia: or The Art of Walking the Streets of London as a mock Georgic deriving its main characteristics and themes from Juvenal's Third Satire and Vi...

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.

For an algebraic number field [Formula: see text], let [Formula: see text] be the associated Dedekind zeta-function. It is conjectured that [Formula: see text] is transcendental for any positive integer [Formula: see text]. The only known case of this conjecture was proved independently by Siegel and Klingen, namely that, when [Formula: see text] is a totally real number field, [Formula: see text] is an algebraic multiple of [Formula: see text] and hence, is transcendental. If [Formula: see text] is not totally real, the question of whether [Formula: see text] is irrational or not remains open. In this paper, we prove that for a fixed integer [Formula: see text], at most one of [Formula: see text] is rational, as [Formula: see text] varies over all imaginary quadratic fields. We also discuss a generalization of this theorem to CM-extensions of number fields.

Let v " be arbitrary. The goal of the present paper is to de-rive isomorphisms. We show that there exists a Dedekind and natural Dirichlet prime. It is essential to consider that g may be tangential. In [43, 10], the authors described elements.

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.

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We explore the connection between atomicity in Prufer domains and their corresponding class groups. We observe that a class group of infinite order is necessary for non-Noetherian almost Dedekind and Prufer domains of finite character to be atomic. We construct a non-Noetherian almost Dedekind domain and exhibit a generating set for the ideal class semigroup.

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.

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.

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.

Dedekind sums occur in the transformation behaviour of the logarithm of the Dedekind eta-function under substitutions from the modular group. In 1892, Dedekind showed a reciprocity relation for the Dedekind sums. Apostol generalized Dedekind sums by replacing the first Bernoulli function appearing in them by any Bernoulli functions and derived a reciprocity relation for the generalized Dedekind sums. In this paper, we consider poly-Dedekind sums which are obtained from the Dedekind sums by replacing the first Bernoulli function by any type 2 poly-Bernoulli functions of arbitrary indices and prove a reciprocity relation for the poly-Dedekind sums., 12 pages

A power series is called lacunary if “almost all” of its coefficients are zero. Integer partitions have motivated the classification of lacunary specializations of Han’s extension of the Nekrasov–Okounkov formula. More precisely, we consider the modular forms $$\begin{aligned}F_{a,b,c}(z) :=\frac{\eta (24az)^a \eta (24acz)^{b-a}}{\eta (24z)},\end{aligned}$$ F a , b , c ( z ) : = η ( 24 a z ) a η ( 24 a c z ) b - a η ( 24 z ) , defined in terms of the Dedekind $$\eta $$ η -function, for integers $$a,c \ge 1$$ a , c ≥ 1 , where $$b \ge 1$$ b ≥ 1 is odd throughout. Serre (Publications Mathématiques de l’IHÉS 123–201:2959–2968, 1981) determined the lacunarity of the series when $$a = c = 1$$ a = c = 1 . Later, Clader et al. (Am Math Soc 137(9):2959–2968, 2009) extended this result by allowing a to be general and completely classified the $$F_{a,b,1}(z)$$ F a , b , 1 ( z ) which are lacunary. Here, we consider all c and show that for $${a \in \{1,2,3\}}$$ a ∈ { 1 , 2 , 3 } , there are infinite families of lacunary series. However, for $$a \ge 4$$ a ≥ 4 , we show that there are finitely many triples (a, b, c) such that $$F_{a,b,c}(z)$$ F a , b , c ( z ) is lacunary. In particular, if $$a \ge 4$$ a ≥ 4 , $$b \ge 7$$ b ≥ 7 , and $$c \ge 2$$ c ≥ 2 , then $$F_{a,b,c}(z)$$ F a , b , c ( z ) is not lacunary. Underlying this result is the proof the t-core partition conjecture proved by Granville and Ono (Trans Am Math Soc 348(1):331–347, 1996).

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\

In this paper, we study the class of prime semimodules and the related concepts, such as the class of semimodules, the class of Dedekind semidomains, the class of prime semimodules which is invariant subsemimodules of its injective hull, and the compressible semimodules. In order to make the work as complete as possible, we stated, and sometimes proved, some known results related to the above concepts.

The aim of this paper is to introduce the concept of Dedekind semimodules and study the related concepts, such as the class of semimodules, and Dedekind multiplication semimodules . And thus study the concept of the embedding of a semimodule in another semimodule.