Let R be an hereditary Noetherian prime ring, let S be a "Dedekind closure" of R and let T be the category of finitely generated S-torsion R-modules. It is shown that for all i ~ 0, there is an exact sequence 0 -> Ki(T) -> Ki(R) -> Ki(S) ...... O. If i = 0, or R has finitely many idempotent ideals then this sequence splits. A notion of ''right ideal class group" is then introduced for hereditar...

In [Ro76], M. Rosen showed that for any countable commutative group G, there is a field K, an elliptic curve E/K and a surjective group homomorphism E(K) → G. From this he deduced that any countable commutative group whatsoever is the ideal class group of an elliptic Dedekind domain – the ring of all functions on an elliptic curve which are regular away from some (fixed, possibly infinite) set ...

Our task here is to recall part of the theory of orders and ideals in quaternion algebras. Some of the theory makes sense in the context of B/K a quaternion algebra over a field K which is the quotient field of a Dedekind ring R. For our purposes K will always be a number field, or the completion of a number field at a finite prime, and R will be the ring of integers of K. (Nevertheless, we sha...

Familiarly, in Z, we have unique factorization. We investigate the general ring and what conditions we can impose on it to necessitate analogs of unique factorization. The trivial ideal structure of a field, the extent to which primary decomposition is unique, that a Noetherian ring necessarily has one, that a principal ideal domain is a unique factorization domain, and that a Dedekind domain h...

A Tychonoff space X such that the maximum ring of quotients of C(X) is uniformly complete is called a uniform quotients space. It is shown that this condition is equivalent to the Dedekind– MacNeille completion of C(X) being a ring of quotients of C(X), in the sense of Utumi. A compact metric space is a uniform quotients space precisely when it has a dense set of isolated points. Extremally dis...

Abstract We show a conditional exactness statement for the Nisnevich Gersten complex associated to an $$\mathbb {A}^1$$ A 1 -invariant cohomology theory with descent smooth schemes over Dedekind ring only infinite residue fields. As application we...

We theoretically study the Kondo effect in a quantum dot embedded in an Aharonov-Bohm ring, using the "poor man's" scaling method. Analytical expressions of the Kondo temperature TK are given as a function of magnetic flux Φ penetrating the ring. In this Kondo problem, there are two characteristic lengths, Lc=ℏvF∕|ε̃0| and LK = ħvF = TK, where vF is the Fermi velocity and ε̃0 is the renormalized ...

In this paper, we establish two results concerning algebraic (C,+)-actions on C. First of all, let φ be an algebraic (C,+)-action on C3. By a result of Miyanishi, its ring of invariants is isomorphic to C[t1, t2]. If f1, f2 generate this ring, the quotient map of φ is the map F : C3 → C2, x 7→ (f1(x), f2(x)). By using some topological arguments, we prove that F is always surjective. Secondly we...

We prove new height inequalities for determinantal ideals in a regular local ring, or more generally in a local ring of given embedding codimension. Our theorems extend and sharpen results of Faltings and Bruns. Introduction. Let φ be a map of vector bundles on a variety X. A wellknown theorem of Eagon and Northcott [EN] gives an upper bound for the codimension of the locus where φ has rank ≤ s...

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.