Proof. We will calculate the integral closure of A inside K. We begin with a more general discussion. Suppose C is an irreducible, affine curve, that is, C is some irreducible affine variety of dimension 1. Let A be the ring of regular functions on C, so that A is a noetherian domain of Krull dimension 1. Localization preserves the noetherian condition, as well as the domain condition. Thus, th...

We introduce elliptic analogues to the Bernoulli ( resp. Euler) numbers and functions. The first aim of this paper is to state and prove that our elliptic Bernoulli and Euler functions satisfied Raabe’s formulas (cf. Theorems 3.1.1, 3.2.1). We define two kinds of elliptic Dedekind-Rademacher sums, in terms of values of our elliptic Bernoulli (resp. Euler) functions. The second aim of this paper...

First, we give a complete description of the indecomposable prime modules over a Dedekind domain. Second, if R is the pullback, in the sense of [9], of two local Dedekind domains then we classify indecomposable prime R-modules and establish a connection between the prime modules and the pure-injective modules (also representable modules) over such rings.

The Abhyankar-Sathaye Problem asks whether any biregular embedding φ : C →֒ C can be rectified, that is, whether there exists an automorphism α ∈ AutC such that α ◦ φ is a linear embedding. Here we study this problem for the embeddings φ : C3 →֒ C4 whose image X = φ(C3) is given in C4 by an equation p = f(x, y)u + g(x, y, z) = 0, where f ∈ C[x, y]\{0} and g ∈ C[x, y, z]. Under certain additional ...

Let G and E stand for one of the following pairs of groups: • Either G is the general quadratic group U(2n, R,Λ), n ≥ 3, and E its elementary subgroup EU(2n, R,Λ), for an almost commutative form ring (R,Λ), • or G is the Chevalley group G(Φ, R) of type Φ, and E its elementary subgroup E(Φ, R), where Φ is a reduced irreducible root system of rank ≥ 2 and R is commutative. Using Bak’s localizatio...

Exercise 10.3.2. Let R be a commutative ring with identity. For all positive integers n and m, R ∼= R if and only if n = m. Proof. Let φ : R → R be an isomorphism of R-modules and let I E R be a maximal ideal. Then the map φ̄ : R → R/IR given by φ̄(α) = φ(α) is a morphism of R-modules. Moreover ker φ̄ = {α ∈ R | φ̄(α) = 0} = {α ∈ R | φ(α) ∈ IR} = φ−1(IRm) = IR. Therefore by the first isomorphism th...

Let k be a field of positive characteristic and K = k(V ) a function field of a variety V over k and let AK be the ring of adéles of K with respect to the places on K corresponding to the divisors on V . Given a Drinfeld module Φ : F[t] → EndK(Ga) over K and a positive integer g we regard both K and AgK as Φ(Fp[t])-modules under the diagonal action induced by Φ. For Γ ⊆ K a finitely generated Φ...

The Abhyankar-Sathaye Problem asks whether any biregular embedding φ : C →֒ C can be rectified, that is, whether there exists an automorphism α ∈ AutC such that α ◦ φ is a linear embedding. Here we study this problem for the embeddings φ : C3 →֒ C4 whose image X = φ(C3) is given in C4 by an equation p = f(x, y)u + g(x, y, z) = 0, where f ∈ C[x, y]\{0} and g ∈ C[x, y, z]. Under certain additional ...

In this work, we study the DNA codes from the ring R = Z4 + wZ4, where w 2 = 2 + 2w with 16 elements. We establish a one to one correspondence between the elements of the ring R and all the DNA codewords of length 2 by defining a distance preserving Gau map φ. Using this map, we give several new classes of the DNA codes which satisfies reverse and reverse complement constraints. Some of the con...