Abstract. Dedekind symbols are generalizations of the classical Dedekind sums (symbols). There is a natural isomorphism between the space of Dedekind symbols with Laurent polynomial reciprocity laws and the space of modular forms. We will define a new elliptic analogue of the Apostol-Dedekind sums. Then we will show that the newly defined sums generate all odd Dedekind symbols with Laurent poly...

The ring of integer-valued polynomials over a given subset S ℤ (or Int(S,ℤ)) is defined as the set in ℚ[x] which maps to ℤ. In factorization theory, it crucial check irreducibility polynomial. this article, we make Bhargava factorials our main tool polynomial f∈Int(S,ℤ)). We also generalize results arbitrary subsets Dedekind domain.

The main result of this paper is a generalization the theorem Chevalley-Shephard-Todd to rings invariants pseudoreflection groups over Dedekind domains. In special case principal ideal domain in which group order invertible it proved that ring isomorphic polynomial ring. An intermediate every finitely generated regular graded algebra tensor product blowup algebras.

For certain classes of Prüfer domains A, we study the completion Â,T ofA with respect to the supremum topology T = sup{Tw|w ∈ Ω}, where Ω is the family of nontrivial valuations on the quotient field which are nonnegative on A and Tw is a topology induced by a valuation w ∈ Ω. It is shown that the concepts ‘SFT Prüfer domain’ and ‘generalized Dedekind domain’ are the same. We show that if E is t...

Let S be a set of n ideals of a commutative ring A and let Geven (respectively Godd) denote the product of all the sums of even (respectively odd) number of ideals of S. If n ≤ 6 the product of Geven and the intersection of all ideals of S is included in Godd. In the case A is an Noetherian integral domain, this inclusion is replaced by equality if and only if A is a Dedekind domain.

In this paper, we introduce the concept of fuzzy star-operations on an integral domain and show that the set of all fuzzy star-operations on the integral domain forms a complete lattice. We also characterize Pr3 ufer domains, psuedo-Dedekind domains, (generalized-) greatest common divisor domains, and other integral domains in terms of the invertibility of certain fractionary fuzzy ideals. c © ...

For a given family (Gi)i∈N of finitely generated abelian groups, we construct Dedekind domain D having the following properties. Pic(D)≅⨁i∈NGi. each i∈N, there exists submonoid Si⊆D• with Pic(DSi)≅Gi. Each class Pic(D) and all Pic(DSi) contains infinitely many prime ideals.

In [6] was proved that if R is a principal ideal domain and N ⊂ M are submodules of R[x1, . . . , xn], then the primary decomposition for N in M can be computed using Gröbner bases. In this paper we extend this result to Dedekind domains. The procedure that computed the primary decomposition is illustrated with an example.

We describe a deterministic polynomial-time test that determining whether a nonzero ideal is a prime ideal in a Dedekind domain with finite rank. The techniques which we used are basis representation of finite rings and the Hermite and Smith normal forms.