Elementary equivalence of some rings of definable functions

We characterize elementary equivalences and inclusions between von Neumann regular real closed rings in terms of their boolean algebras of idempotents, and prove that their theories are always decidable. We then show that, under some hypotheses, the map sending an L-structure R to the L-structure of definable functions from R ⁿ to R preserves elementary inclusions and equivalences and gives a structure with a decidable theory whenever R is decidable. We briefly consider structures of definable functions satisfying an extra condition such as continuity.      

Rings of quotients of C(X) induced by points

We study two rings of quotients of C(X), the ring of continuous functions on the Tychonoff space X. The first, F(X), is the ring of quotients induced by the filter of ideals consisting of dense finite intersections of fixed maximal ideals. The second, C[F], is the ring of quotients induced by the filter of dense cofinite subspaces of X. After some preliminary information we explicitly describe in §2 and §3 the constructions of these rings of quotients. In the third section, we use F(X)and C[F] to define and study the class of h-points and h-spaces. In particular, we show that C-spaces and P-spaces are h-spaces. In the last section we construct an ideal of C(X)which will be used to give an ideal theoretic characterization of h-points. This revised version was published online in June 2006 with corrections to the Cover Date.     

Essential Ideals in Rings of Measurable Functions

In this article, we study essential ideals, socle, and some related ideals of rings of real valued measurable functions. We also study the Goldie dimension of rings of measurable functions.     

An Example of Bergman's and the Extension Problem for Clean Rings

An example of Bergman is used to show that the extension of a clean ring by another clean ring need not be clean. That is, there exists a ring R and an ideal I of R such that both R/I and I are clean and idempotents lift modulo I, but R is not clean.     

Conditions on Normal Spaces of Finite Rank That Guarantee C(X) is SV

An f-ring A is an SV f-ring if for every minimal prime ?-ideal P of A, A/P is a valuation domain. A topological space X is an SV space if C(X) is an SV f-ring. For normal spaces, several conditions are shown to guarantee the space is an SV space. For example, a normal space of finite rank for which the closure of the set of points of rank greater than 1 is an F-subspace, is an SV space. For normal spaces of rank 2, a characterization of SV spaces is given.     

Integrally Closed Domains,Minimal Polynomials,and Null Ideals of Matrices

Abstract

We show that every element of the integral closure D′ of a domain D occurs as a coefficient of the minimal polynomial of a matrix with entries in D. This answers affirmatively a question of Brewer and Richman, namely, if integrally closed domains are characterized by the property that the minimal polynomial of every square matrix with entries in D is in D[x]. It follows that a domain D is integrally closed if and only if for every matrix A with entries in D the null ideal of A, N _D (A)?=?{f?∈?D[x]?∣?f(A)?=?0} is a principal ideal of D[x].     

Power Residue Symbols and the Central Sections of SL(2, A)

In TheE(2,A) sections of SL(2,A) (Ann. of Math. 134 (1991), 159–188), we locate the E(A) normalized subgroups of SL(2,A) in central sections of SL(2,A) for all subrings of Q and all commutative rings satisfying SR ₂ In solving this problem we introduced the notion of radix (see (1.1)) and the group C(P_x) = [E(2,A),E(2,A;P_x)] = [SL(2,A), SL(2,A;P_x)] for the rings considered here.The purpose of this paper is to determine SL(2,A;P_xC(P_x) for SR ₂ rings and number rings with infinitely many units.In Section 2, Mennicke symbols for Jordan ideals are defined. They are determined for number rings and shown to be connected to power residue symbols in a delicate way. This extends the work of Bass, Milnor and Serre.In Section 3, an explicit homomorphism from E(2,Al;P_x) into an additive section of A is given for all commutative rings A. If A satisfiesSR ₂ the kernel of this map is C(P_x.The main problem for number rings is solved by giving an explicit homomorphism on SL(2,A;P_x) whose kernel is C(P_x).     

A(X,M)-c-内射性

本文研究了X-扩张模.利用A( X,M)-c-内射模的概念,获得了A( X,M)-c-内射模的有限直和仍是A( X,M)-c-内射模的一个充分必要条件,并推广了相应的结果.     

A System of Four Matrix Equations over von Neumann Regular Rings and Its Applications

We consider the system of four linear matrix equations A1X = C1, XB2=C2, A3XB3=C3 and A4XB4 = C4 over h, an arbitrary von Neumann regular ring with identity. A necessary and sufficient condition for the existence and the expression of the general solution to the system are derived. As applications, necessary and sufficient conditions are given for the system of matrix equations A1X = C1 and A3X=C3 to have a bisymmetric solution, the system of matrix equations A1X = C1 and A3XB3 = C3 to have a perselfconjugate solution over h with an involution and char h≠2, respectively. The representations of such solutions are also presented. Moreover, some auxiliary resultson other systems over h are obtained. The previous known results on some systems of matrix equations are special cases of the new results.     

Integral Multilinear Forms on C(K, X) Spaces

We use polymeasures to characterize when a multilinear form defined on a product of C(K, X) spaces is integral.     

(0,M)插值的推广

本文对周期为2ψ的函数引入了一类新的函数插值并建立了插值的收敛速度估计．通常的Birkhoff插值是该插值的极限情形．     

On an Algebraic Extension of A(E)

An algebraic extension of the algebra A(E), where E is a compactum in with nonempty connected interior, leads to a Banach algebra B of functions that are holomorphic on some analytic set K^{o 2} with boundary bK and continuous up to bK. The singular points of the spectrum of B and their defects are investigated. For the case in which B is a uniform algebra, the depth of B in the algebra C(bK) is estimated. In particular, conditions under which B is maximal on bK are obtained.     

Deza graphs with parameters (v,k,k−2,a)

A Deza graph with parameters $\left(v,k,b,a\right)$ is a $k$‐regular graph on $v$ vertices in which the number of common neighbors of two distinct vertices takes one of the following values: $b$ or $a$, where $b\ge a$. In the previous papers Deza graphs with $b=k?1$ were characterized. In this paper, we characterize Deza graphs with $b=k?2$.     

A Note on Bruhat Decomposition of GL(n) over Local Principal Ideal Rings

Let A be a local commutative principal ideal ring. We study the double coset space of GL _n (A) with respect to the subgroup of upper triangular matrices. Geometrically, these cosets describe the relative position of two full flags of free primitive submodules of A ⁿ . We introduce some invariants of the double cosets. If k is the length of the ring, we determine for which of the pairs (n,k) the double coset space depends on the ring in question. For n = 3, we give a complete parametrisation of the double coset space and provide estimates on the rate of growth of the number of double cosets.     

A Classification of Regular p-Groups with Invariants (e, 2, 1)

The class of the regular p-groups is one of the important classes in p-groups. Not only it has many similar properties as abelian p-groups, but also many of the p-groups belong to this class. In this paper, using the algorithms for determining the isomorphic regular p-groups, we give a complete classification of the regular p-groups with e-invariants (e, 2, 1).Supported by SXYSF 991003.     

Dual representations of hulls for functions satisfying f (0) = inf f (X\{0}) *

Let Xbe a setW set of extended-real valued functions on Xand 0 an element of Xsuch that w(0) = 0 for all win W. We give a general theory of dual representations, without any extra parameters, of various hulls for extended-real valued functions on X satisfying f(0)= inf f(X\{0})We use the unifying framework of quasi-convex functions with respect to families of subsets of Xand a condition generalizing the bipolar theorem. Our results contain, as particular cases, some recent results of Thach (1991, 1993) and Rubinov and Glover (1996) on W-pseudo-affine and W-quasi-coaffine hulls.We also give some

results in the converse direction. These yield, in particular, that the bipolar theorem is equivalent to a certain property of the lower semi-continuous quasi-convex hulls of functions on locally convex spaces     

De Morgan (2, n)-Semigroups of n-Place Functions

An abstract characterization of sets of some partial functions from Aⁿ to A closed with respect to the Menger's and Mann's superpositions and a quasicomplementation is presented.