Embedding Witt rings of Dedekind domains

Let W(R) denote Harrison's Witt ring of the commutative ring R. In case R is a field of characteristic ≠ 2, this is the classical Witt ring based on anisotropic quadratic forms. In this note we determine under what conditions W(R) is embedded in W(S) for certain Dedekind domains R ? S. In particular, an answer is given in case R and S are the integers in algebraic number fields K and L, respectively, with (L: K) odd.     

Decomposition of integer-valued polynomial algebras

Let D be a commutative domain with field of fractions K, let A be a torsion-free D-algebra, and let B be the extension of A to a K-algebra. The set of integer-valued polynomials on A is $\mathrm{Int}(A)=\{f\in B[X]|f(A)?A\}$, and the intersection of $\mathrm{Int}(A)$ with $K[X]$ is ${\mathrm{Int}}_K(A)$, which is a commutative subring of $K[X]$. The set $\mathrm{Int}(A)$ may or may not be a ring, but it always has the structure of a left ${\mathrm{Int}}_K(A)$-module.A D-algebra A which is free as a D-module and of finite rank is called ${\mathrm{Int}}_K$-decomposable if a D-module basis for A is also an ${\mathrm{Int}}_K(A)$-module basis for $\mathrm{Int}(A)$; in other words, if $\mathrm{Int}(A)$ can be generated by ${\mathrm{Int}}_K(A)$ and A. A classification of such algebras has been given when D is a Dedekind domain with finite residue rings. In the present article, we modify the definition of ${\mathrm{Int}}_K$-decomposable so that it can be applied to D-algebras that are not necessarily free by defining A to be ${\mathrm{Int}}_K$-decomposable when $\mathrm{Int}(A)$ is isomorphic to ${\mathrm{Int}}_K(A){?}_{D}A$. We then provide multiple characterizations of such algebras in the case where D is a discrete valuation ring or a Dedekind domain with finite residue rings. In particular, if D is the ring of integers of a number field K, we show that an ${\mathrm{Int}}_K$-decomposable algebra A must be a maximal D-order in a separable K-algebra B, whose simple components have as center the same finite unramified Galois extension F of K and are unramified at each finite place of F. Finally, when both D and A are rings of integers in number fields, we prove that ${\mathrm{Int}}_K$-decomposable algebras correspond to unramified Galois extensions of K.     

On the Brauer group and factoriality of normal domains

Let B(R) be the Brauer group of the integrally closed noetherian domain R with quotient field K. We reexamine the proof that B(R) → R(K) is monic for R regular from the point of view of factoriality of R and its extensions. For R local with maximal ideal m, henselization R^h and divisor class group Cl(R) we embed ker $\text{{B(R) → B(K)?B(}\text{R}\text{M}\text{)}}$ into $\text{Cl(R}{}^{\mathrm{h}}\text{)}\text{Cl(R)}$. This is applied to obtain examples of non-regular geometric local domains R for which B(R) → B(K) is monic.     

Henriksen?s contributions to residue class rings of analytic and entire functions

The author surveys, summarizes and generalizes results of Golasiński and Henriksen, and of others, concerning certain residue class rings.Let A(R) denote the ring of analytic functions over reals R and E(K) the ring of entire functions over R or complex numbers C. It is shown that if m is a maximal ideal of A(R), then A(R)/m is isomorphic either to the reals or a real-closed field that is η₁-set, while if m is a maximal ideal of E(K), then E(K)/m is isomorphic to one of these latter two fields or to complex numbers.     

Strong cleanness of generalized matrix rings over a local ring

Let K_s(R) be the generalized matrix ring over a ring R with multiplier s. For a general local ring R and a central element s in the Jacobson radical of R, necessary and sufficient conditions are obtained for K_s(R) to be a strongly clean ring. For a commutative local ring R and an arbitrary element s in R, criteria are obtained for a single element of K_s(R) to be strongly clean and, respectively, for the ring K_s(R) to be strongly clean. Specializing to s = 1 yields some known results. New families of strongly clean rings are presented.     

Uppers to zero in R[x] and almost principal ideals

Let R be an integral domain with quotient field K and f(x) a polynomial of positive degree in K[x]. In this paper we develop a method for studying almost principal uppers to zero ideals. More precisely, we prove that uppers to zero divisorial ideals of the form I = f(x)K[x] ∩ R[x] are almost principal in the following two cases:

• J, the ideal generated by the leading coefficients of I, satisfies J ^?1 = R.
• I ^?1 as the R[x]-submodule of K(x) is of finite type.

Furthermore we prove that for I = f(x)K[x] ∩ R[x] we have:

• I ^?1 ∩ K[x] = (I: K(x) I).
• If there exists p/q ∈ I ^?1 ? K[x], then (q, f) ≠ 1 in K[x]. If in addition q is irreducible and I is almost principal, then I′ = q(x)K[x] ∩ R[x] is an almost principal upper to zero.

Finally we show that a Schreier domain R is a greatest common divisor domain if and only if every upper to zero in R[x] contains a primitive polynomial.     

Leavitt path algebras with coefficients in a commutative ring

Given a directed graph E we describe a method for constructing a Leavitt path algebra L_R(E) whose coefficients are in a commutative unital ring R. We prove versions of the Graded Uniqueness Theorem and Cuntz-Krieger Uniqueness Theorem for these Leavitt path algebras, giving proofs that both generalize and simplify the classical results for Leavitt path algebras over fields. We also analyze the ideal structure of L_R(E), and we prove that if K is a field, then L_K(E)≅K⊗_ZL_Z(E).     

Asymmetry of twisted convolution operators

Let K be a distribution on $\text{R}$². We denote by λ(K) the twisted convolution operator f → K × f defined by the formula K × f(x, y) = ∝∝ dudvK(x ? u, y ? v) f(u, v) exp(ixv ? iyu). We show that there exists K such that the operator λ(K) is bounded on L^p(R)² for every p in (1, 2¦, but is unbounded on L^q(R)² for every q > 2.     

Lifting Artin–Schreier covers with maximal wild monodromy

Let k be an algebraically closed field of characteristic p > 0. We consider the problem of lifting p-cyclic covers of ${\mathbb{P}^{1}_k}$ as p-cyclic covers C of the projective line over some discrete valuation field K under the condition that the wild monodromy is maximal. We answer positively the problem for covers birationally given by w ^p ?w = t R(t) for any additive polynomial R(t). One gives further informations about the ramification filtration of the monodromy extension and in the case when p = 2, one computes the conductor exponent f (Jac(C)/K) and the Swan conductor sw(Jac(C)/K).     

On the minimum size of binary codes with length 2<Emphasis Type="Italic">R</Emphasis> + 4 and covering radius <Emphasis Type="Italic">R</Emphasis>

The minimum size of a binary code with length n and covering radius R is denoted by K(n, R). For arbitrary R, the value of K(n, R) is known when n ≤  2R +  3, and the corresponding optimal codes have been classified up to equivalence. By combining combinatorial and computational methods, several results for the first open case, K(2R +  4, R), are here obtained, including a proof that K(10, 3) =  12 with 11481 inequivalent optimal codes and a proof that if K(2R +  4, R) <  12 for some R then this inequality cannot be established by the existence of a corresponding self-complementary code.     

Metrizability of Spaces of ℝ-places of Function Fields of Transcendence Degree 1 Over Real Closed Fields

In this article we discuss the following question “When do different orderings of the rational function field R(X) (where R is a real closed field) induce the same ?-place?” We use this to show that if R contains a dense real closed subfield R′, then the spaces of ?-places of R(X) and R′(X) are homeomorphic. For the function field K = R(X), we prove that its space M(K) of ?-places is metrizible if and only if R contains a countable dense subfield. Moreover, we show that this condition is neccessary for the metrizability of M(F) for any function field F of transcendence degree 1 over R.     

Separable subspaces of affine function spaces on convex compact sets

Let K be a compact convex subset of a separated locally convex space (over R) and let Ap(K) denote the space of all continuous real-valued affine mappings defined on K, endowed with the topology of pointwise convergence on the extreme points of K. In this paper we shall examine some topological properties of Ap(K). For example, we shall consider when Ap(K) is monolithic and when separable compact subsets of Ap(K) are metrizable.     

Existence Criterion for Estimates of Derivatives of Rational Functions

Suppose that K is a compact set in the open complex plane. In this paper, we prove an existence criterion for an estimate of Markov-Bernstein type for derivatives of a rational function R(z) at any fixed point z ₀ ∈ K. We prove that, for a fixed integer s, the estimate of the form |R ^(s) (z ₀)| ≤ C(K, z ₀, s)n‖R‖ _C(K), where R is an arbitrary rational function of degree n without poles on K and C is a bounded function depending on three arguments K, z ₀, and s, holds if and only if the supremum $$\omega (K,z_0 ,s) = \sup \left\{ {\frac{{\operatorname{dist} (z,K)}}{{\left| {z - z_0 } \right|^{s + 1} }}} \right\}$$ over z in the complement of K is finite. Under this assumption, C is less than or equal to const ·s!ω(K, z ₀, s).     

On mappings related to the gradient of the conformal radius

We establish a criterion for the gradient ?R(D, z) of the conformal radius of a convex domain D to be conformal: the boundary ?D must be a circle. We obtain estimates for the coefficients K(r) for the K(r)-quasiconformal mappings ?R(D, z), D(r) ? D, 0 < r < 1, and supplement the results of Avkhadiev and Wirths concerning the structure of the boundary under diffeomorphic mappings of the domain D.     

A field-theoretic invariant for domains

SoientR ?T des anneaux intègres. D’après Dobbs-Mullins, on pose Λ(T/R) ? sup{λ(k _Q (T)/k _Q∩R (R)) |Q ∈ Spec(T)} où, pour des corpsK?L,λ(L/K) est la longueur maximale d’une chaîne de corps contenus entreK etL. On introduitσ(R):=sup{Λ(T/R)|T est un suranneau deR\. On détermineσ(R) siR′, la clôture intégrale deR, est un anneau de Prüfer et également siR est un anneau de pseudo-valuation. On considère le cas oùσ(R)=1, en particulier siR′ est une extension minimale deR. Plusieurs calculs sont facilités par un résultat sur les carrés cartésiens, et il y a des exemples divers.     

Higher Algebraic <Emphasis Type="Italic">K</Emphasis>-theory of Ring Epimorphisms

Given a homological ring epimorphism from a ring R to another ring S, we show that if the left R-module S has a finite-type resolution, then the algebraic K-group K _n (R) of R splits as the direct sum of the algebraic K-group K _n (S) of S and the algebraic K-group K _n (R) of a Waldhausen category R determined by the ring epimorphism. This result is then applied to endomorphism rings, matrix subrings, rings with idempotent ideals, and universal localizations which appear often in representation theory and algebraic topology.     

<Emphasis Type="Bold">K</Emphasis><Subscript>0</Subscript> of Semiartinian Von Neumann Regular Rings. Direct Finiteness Versus Unit-Regularity

If R is a regular and semiartinian ring, it is proved that the following conditions are equivalent: (1) R is unit-regular, (2) every factor ring of R is directly finite, (3) the abelian group K _O(R) is free and admits a basis which is in a canonical one to one correspondence with a set of representatives of simple right R-modules. For the class of semiartinian and unit-regular rings the canonical partial order of K _O(R) is investigated. Starting from any partially ordered set I, a special dimension group G(I) is built and a large class of semiartinian and unit-regular rings is shown to have the corresponding K _O(R) order isomorphic to G(P r i m _R ), where P r i m _R is the primitive spectrum of R. Conversely, if I is an artinian partially ordered set having a finite cofinal subset, it is proved that the dimension group G(I) is realizable as K _O(R) for a suitable semiartinian and unit-regular ring R.     

Int-decomposable algebras

Let D be a Dedekind domain with fraction field k. Let A be a D-algebra that, as a D-module, is free of finite rank. Let B be the extension of A to a k-algebra. The set of integer-valued polynomials over A   is defined to be Int(A)={f∈B[x]|f(A)⊆A}$\mathrm{Int}(A)=\{f\in B[x]|f(A)\subseteq A\}$. Restricting the coefficients to elements of k  , we obtain the commutative ring _Intk(A)={f∈k[x]|f(A)⊆A}${\mathrm{Int}}_k(A)=\{f\in k[x]|f(A)\subseteq A\}$; this makes Int(A)$\mathrm{Int}(A)$ a left _Intk(A)${\mathrm{Int}}_k(A)$-module. Previous researchers have noted instances when a D-module basis for A   is also an _Intk(A)${\mathrm{Int}}_k(A)$-basis for Int(A)$\mathrm{Int}(A)$. We classify all the D-algebras A   with this property. Along the way, we prove results regarding Int(A)$\mathrm{Int}(A)$, its localizations at primes of D, and finite residue rings of A.     

When is c(x) a Clean Ring?

An element of a ring R is called clean if it is the sum of a unit and an idempotent and a subset A of R is called clean if every element of A is clean. A topological characterization of clean elements of C(X) is given and it is shown that C(X) is clean if and only if X is strongly zero-dimensional, if and only if there exists a clean prime ideal in C(X). We will also characterize topological spaces X for which the ideal C_K(X) is clean. Whenever X is locally compact, it is shown that C_K(X) is clean if and only if X is zero-dimensional.     

Elasticity Properties Preserved in the Normset

Let T⊆R be the rings of integers in a number field and a finite Galois extension field. We study relations between the elasticity ρ(R) of the monoid of nonzero elements of R and the elasticity ρ(S) of the monoid S of norms to T of those elements. We show ρ(R)?ρ(S) and that equality holds if the norms of irreducible elements of R are irreducible in S, which is true, in particular, if either ρ(R)<2 or ρ(S)=1.