We recall that Theorem 1.3 allows us to define the ideal class group of a Dedekind domain, and in particular of a ring of integers, as the group of fractional ideals modulo the subgroup of principal ideals. We will prove that in the case of a ring of integers, the ideal class group is finite. In fact, we will shortly give a stronger statement due to Minkowski. Using similar techniques, we will ...

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.

Let R be a ring and S a nonempty subset of R. Suppose that θ and φ are endomorphisms of R. An additive mapping δ : R → R is called a left (θ,φ)-derivation (resp., Jordan left (θ,φ)derivation) on S if δ(xy) = θ(x)δ(y)+φ(y)δ(x) (resp., δ(x2) = θ(x)δ(x)+φ(x)δ(x)) holds for all x,y ∈ S. Suppose that J is a Jordan ideal and a subring of a 2-torsion-free prime ring R. In the present paper, it is show...

In [5], we computed the cyclic homology of finite extensions of Dedekind domains. The crucial result was a formula for the cyclic homology, relative to a ground ring R, of A = R[x]/(P (x)). This paper presents two extensions of the methods and results of this paper. Recall that Hochschild homology, though generally defined in terms of a bar complex, is almost never, in practice, computed from i...

A theorem of B. Green states that if A is a Dedekind ring whose fraction field is a local or global field, every normal projective curve over Spec(A) has a finite morphism to P1A. We give a different proof of a variant of this result using intersection theory and work of Moret-Bailly.

In this note we extend duality theorems for crossed products obtained by M. Koppinen and C. Chen from the case of a base field or a Dedekind domain to the case of an arbitrary noetherian commutative ground ring under fairly weak conditions. In particular we extend an improved version of the celebrated Blattner-Montgomery duality theorem to the case of arbitrary noetherian ground rings.

Let R = D[x;σ, δ] be an Ore extension over a commutative Dedekind domain D, where σ is an automorphism on D. In the case δ = 0 Marubayashi et. al. already investigated the class of minimal prime ideals in term of their contraction on the coefficient ring D. In this note we extend this result to a general case δ 6= 0.

Richard Dedekind's characterization of the real numbers as the system of cuts of rational numbers is by now the standard in almost every mathematical book on analysis or number theory. In the philosophy of mathematics Dedekind is given credit for this achievement, but his more general views are discussed very rarely and only superrcially. For example, Leo Corry, who dedicates a whole chapter of...