We study the relation of the ideal class group of a Dedekind domain A to that of As, where S is a multiplicatively closed subset of A. We construct examples of (a) a Dedekind domain with no principal prime ideal and (b) a Dedekind domain which is not the integral closure of a principal ideal domain. We also obtain some qualitative information on the number of non-principal prime ideals in an ar...

Let φ be a w-continuous homomorphism from a dual Banach algebra to C. The notion of φ-Connes amenability is studied and some characterizations is given. A type of diagonal for dual Banach algebras is dened. It is proved that the existence of such a diagonal is equivalent to φ-Connes amenability. It is also shown that φ-Connes amenability is equivalent to so-called φ-splitting of a certain short...

1. Concerning These Notes 2 2. Introduction (First Day) 2 3. Algebraic Numbers and Integers 5 4. Linear Algebra for Number Theory 10 5. Review of Field Theory; Traces/Norms 10 6. Units: Some Notes Missing Here 16 7. Diophantine Approximation 16 8. The Trace Pairing 18 9. Linear Algebra and Discriminants 21 10. The Ring of Integers Inside the Number Field 26 11. Some Computational Aspects of Dis...

We introduce the notion of a matroid M over a commutative ring R, assigning to every subset of the ground set an R-module according to some axioms. When R is a field, we recover matroids. When R = Z, and when R is a DVR, we get (structures which contain all the data of) quasi-arithmetic matroids, and valuated matroids, respectively. More generally, whenever R is a Dedekind domain, we extend the...

We consider homogeneous varieties of linear algebras over an associative-commutative ring K with 1, i.e., the varieties in which free algebras are graded. Let F = F (x1, ..., xn) be a free algebra of some variety Θ of linear algebras over K freely generated by a set X = {x1, ..., xn}, EndF be the semigroup of endomorphisms of F , and Aut EndF be the group of automorphisms of the semigroup EndF ...

It is well-known that, given a Dedekind category R the category of (typed) matrices with coeecients from R is a Dedekind category with arbitrary relational sums. In this paper we show that under slightly stronger assumptions the converse is also true. Every atomic Dedekind category R with relational sums and subobjects is equivalent to a category of matrices over a suitable basis. This basis is...

For given N=pq with p and q different odd primes and natural m Banghe Li introduced the public key cryptosystem [1]. In the case m=1 the system is just the famous RSA public key cryptosystem [2]. The cryptosystem is more secure in general [2] than RSA system. But one has to solve a few problems connected with the introduced cryptosystem. The cryptosystem works with elements of the quotient ring...

Let R be a regular local ring, K its field of fractions and (V, φ) a quadratic space over R. In the case of R containing a field of characteristic zero we show that if (V, φ)⊗R K is isotropic over K, then (V, φ) is isotropic over R. 1 Characteristic zero case 1.0.1 Theorem (Main). Let R be a regular local ring, K its field of fractions and (V, φ) a quadratic space over R. Suppose R contains a f...

The theorem in question gives a sufficient condition for a quadratic space over a local ring R to contain a hyperbolic plane over R. Our main aim in this paper is to prove the following result, which is a variant of the Springer theorem [La] for quadratic spaces over local rings. Theorem. Let R be a local Noetherian domain that has an infinite residue field of characteristic different from 2. L...