Quasi-duality for the rings of generalized power series *

Let A, B be associative rings with identity, and (S, ≤) a strictly totally ordered commutative monoid which is also artinian. For any bimodule _AM_B , we construct a bimodule _A[[S]]M[S]_B[[S]] and prove that _AM_B defines a quasi-duality if and only if the bimodule _A[[S]]M[S]_B[[S]] defines a quasi-duality. As a corollary, it is shown that if a ring A has a quasi-duality then the ring A[[S]] of generalized power series over A has a quasi-duality.     

t-closed rings of formal power series

Let A ì BA\subset B be rings. We say that A is t-closed in B, if for each a ? Aa\in A and b ? Bb\in B such that b²-ab,b³-ab² ? Ab^2-ab,b^3-ab^2\in A, then b ? Ab\in A. We present a sufficient condition for the ring A[[X₁,?,X_n]]A[[X_1,\ldots ,X_n]] to be t-closed in B[[X₁,?,X_n]]B[[X_1,\ldots ,X_n]]. By an example, we show that our condition is not necessary. Even though the question is still open, some important cases are treated. For example, if A ì BA\subset B is an integral extension, or if A is p-injective, then A[[X₁,?,X_n]]A[[X_1,\ldots ,X_n]] is t-closed in B[[X₁,?,X_n]]B[[X_1,\ldots ,X_n]] if and only if A is t-closed in B.     

Triangular Matrix Representations of Rings of Generalized Power Series

Let R be a ring and S a cancellative and torsion-free monoid and 〈 a strict order on S. If either （S,≤） satisfies the condition that 0 ≤ s for all s ∈ S, or R is reduced, then the ring [[R^S,≤]] of the generalized power series with coefficients in R and exponents in S has the same triangulating dimension as R. Furthermore, if R is a PWP ring, then so is [[R^S,≤]].     

PP-Rings of Generalized Power Series

Abstract As a generalization of power series rings, Ribenboim introduced the notion of the rings of generalized power series. Let R be a commutative ring, and (S, ≤) a strictly totally ordered monoid. We prove that (1) the ring [[R ^S,≤]] of generalized power series is a PP-ring if and only if R is a PP-ring and every S-indexed subset C of B(R) (the set of all idempotents of R) has a least upper bound in B(R) and (2) if (S, ≤) also satisfies the condition that 0 ≤s for any s∈S, then the ring [[R ^S,≤ ]] is weakly PP if and only if R is weakly PP. Research supported by National Natural Science Foundation of China, 19501007, and Natural Science Foundation of Gansu, ZQ-96-01     

Morita Duality for the Rings of Generalized Power Series

Let A, B be associative rings with identity, and (S, ≤) a strictly totally ordered monoid which is also artinian and finitely generated. For any bimodule _A M _B , we show that the bimodule _{[[ AS,≤ ]]}[M ^{S ,≤}]_{[[ BS, ≤ ]]} defines a Morita duality if and only if _A M _B defines a Morita duality and A is left noetherian, B is right noetherian. As a corollary, it is shown that the ring [[A ^{S ,≤}]] of generalized power series over A has a Morita duality if and only if A is a left noetherian ring with a Morita duality induced by a bimodule _A M _B such that B is right noetherian. Received April 13, 1999, Accepted December 12, 1999     

Special Properties of Rings of Generalized Power Series

Abstract

Let R be a ring and (S, ≤) a strictly ordered monoid. Properties of the ring [[R ^S,≤]] of generalized power series with coefficients in R and exponents in S are considered in this paper. It is shown that [[R ^S,≤]] is reduced (2-primal, Dedekind finite, clean, uniquely clean) if and only if R is reduced (2-primal, Dedekind finite, clean, uniquely clean, respectively) under some additional conditions. Also a necessary and sufficient condition is given for rings under which the ring [[R ^S,≤]] is a reduced left PP-ring.     

Quasi-hopf algebra actions and smash products

A classical result of Magnus asserts that a free group F has a faithful representation in the group of units of a ring of non-commuting formal power series with integral coefficients. We generalize this result to the category of A-groups, where A is an associative ring or an Abelian group. We say that a free A-group F^A has a faithful Magnus representation if there is a ring B containing A as an additive subgroup (or a subring) such that F^A is faithfully represented (exactly as in Magnus' classical result in the case A = Z)in the group of units of the ring of formal power series in non-communting indeterminater over B.The three principal results are: (I) If A is a torsion free Abelian group and F^A is a free A-groupp of Lyndon' type, then F^A has a faithful Magnus representation; (II) If A is an ω‐residually Z ring, then F^A has a faithful Magnus representation;(III) for every nontrivial torsion-free Abelian group A, F^A has a faithful Magnus representation in B[[Y]] for a suitable ring B in and only if F^Q has a faithful Magnus representation in Q[[Y]].     

Special isomorphisms of F[x 1,..., x n ] preserving GCD and their use

On the ring R = F[x ₁,..., x _n ] of polynomials in n variables over a field F special isomorphisms A’s of R into R are defined which preserve the greatest common divisor of two polynomials. The ring R is extended to the ring S: = F[[x ₁,..., x _n ]]⁺ and the ring T: = F[[x ₁,..., x _n ]] of generalized polynomials in such a way that the exponents of the variables are non-negative rational numbers and rational numbers, respectively. The isomorphisms A’s are extended to automorphisms B’s of the ring S. Using the property that the isomorphisms A’s preserve GCD it is shown that any pair of generalized polynomials from S has the greatest common divisor and the automorphisms B’s preserve GCD. On the basis of this Theorem it is proved that any pair of generalized polynomials from the ring T = F[[x ₁,..., x _n ]] has a greatest common divisor.     

The Ascending Chain Condition for Principal Ideals of Rings of Generalized Power Series

Abstract

Let (S, ≤) be a strictly totally ordered monoid and R a domain. It is shown in this paper that [[R ^S,≤]], the ring of generalized power series with coefficients in R and exponents in S, satisfies the ascending chain condition for principal ideals if and only if the ring R and the monoid S satisfy the ascending chain condition for principal ideals of R, and of S, respectively.     

The combinatorial interpretation of the Jacobi identity from Lie algebras

Many enumeration problems concerning sequences emerge as special cases of the combinatorial interpretation of the identity tr log(I ? B) = log det(I ? B), where B is a matrix over the ring of formal power series with zero constant term. The identity is obtained from the well-known formula of Jacobi, namely, det(e^A) = e^tr(A). The problems include the derangement problem, the Simon Newcomb problem, the Smirnov problem, the Ménage problem, and their various generalizations. A conjecture is obtained for the enumeration of sequences with no increasing p runs.     

Semisimilarity for matrices over a division ring

Two square matrices A and B over a ring R are semisimilar, written A$\text{?}$B, if YAX=B and XBY=A for some (possibly rectangular) matrices X, Y over R. We show that if A and B have the same dimension, and if the ring is a division ring $\text{D}$, then A$\text{?}$B if and only if A² is similar to B² and rank(A^k)=rank(B^k), k=1,2,…     

Interactions

Given a C^∗-algebra B, a closed ^*-subalgebra A⊆B, and a partial isometry S in B which interacts with A in the sense that S^∗aS=H(a)S^∗S and SaS^∗=V(a)SS^∗, where V and H are positive linear operators on A, we derive a few properties which V and H are forced to satisfy. Removing B and S from the picture we define an interaction as being a pair of maps (V,H) satisfying the derived properties. Starting with an abstract interaction (V,H) over a C^∗-algebra A we construct a C^∗-algebra B containing A and a partial isometry S whose interaction with A follows the above rules. We then discuss the possibility of constructing a covariance algebra from an interaction. This turns out to require a generalization of the notion of correspondences (also known as Pimsner bimodules) which we call a generalized correspondence. Such an object should be seen as an usual correspondence, except that the inner-products need not lie in the coefficient algebra. The covariance algebra is then defined using a natural generalization of Pimsner's construction of the celebrated Cuntz-Pimsner algebras.     

The Lyapunov matrix equation SA+A1S=S1B1BS

The matrix equation $\text{SA+A}{}^1\text{S=S}{}^1\text{B}{}^1\text{BS}$ is studied, under the assumption that (A, B¹) is controllable, but allowing nonhermitian S. An inequality is given relating the dimensions of the eigenspaces of A and of the null space of S. In particular, if B has rank 1 and S is nonsingular, then S is hermitian, and the inertias of A and S are equal. Other inertial results are obtained, the role of the controllability of (A¹, B¹S¹) is studied, and a class of D-stable matrices is determined.     

Some inequalities for the rational power of a nonnegative definite matrix

In this note the author gives a simple proof of the following fact: Let r and s be two positive rational numbers such that r ? s and let A and B be two n × n non-negative definite Hermitian matrices such that A^r ? B^r. Then A^S ? B^s.     

Noisy colored point set matching

We propose a process for determining approximated matches, in terms of the bottleneck distance, under color preserving rigid motions, between two colored point sets A,B∈R², |A|≤|B|. We solve the matching problem by generating all representative motions that bring A close to a subset B^′ of set B and then using a graph matching algorithm. We also present an approximate matching algorithm with improved computational time. In order to get better running times for both algorithms we present a lossless filtering preprocessing step. By using it, we determine some candidate zones which are regions that contain a subset S of B such that A may match one or more subsets B^′ of S. Then, we solve the matching problem between A and every candidate zone. Experimental results using both synthetic and real data are reported to prove the effectiveness of the proposed approach.     

Classifying simple closed curve pairs in the 2-sphere and a generalization of the Schoenflies theorem

A classification theory is developed for pairs of simple closed curves (A,B) in the sphere S², assuming that A∩B has finitely many components. Such a pair of simple closed curves is called an SCC-pair, and two SCC-pairs (A,B) and (A^′,B^′) are equivalent if there is a homeomorphism from S² to itself sending A to A^′ and B to B^′. The simple cases where A and B coincide or A and B are disjoint are easily handled. The component code is defined to provide a classification of all of the other possibilities. The component code is not uniquely determined for a given SCC-pair, but it is straightforward that it is an invariant; i.e., that if (A,B) and (A^′,B^′) are equivalent and C is a component code for (A,B), then C is a component code for (A^′,B^′) as well. It is proved that the component code is a classifying invariant in the sense that if two SCC-pairs have a component code in common, then the SCC-pairs are equivalent. Furthermore code transformations on component codes are defined so that if one component code is known for a particular SCC-pair, then all other component codes for the SCC-pair can be determined via code transformations. This provides a notion of equivalence for component codes; specifically, two component codes are equivalent if there is a code transformation mapping one to the other. The main result of the paper asserts that if C and C^′ are component codes for SCC-pairs (A,B) and (A^′,B^′), respectively, then (A,B) and (A^′,B^′) are equivalent if and only if C and C^′ are equivalent. Finally, a generalization of the Schoenflies theorem to SCC-pairs is presented.     

Additive results for the Wg-Drazin inverse

In this paper we prove the formula for the expression (A+B)^d,W in terms of A,B,W,A^d,W,B^d,W, assuming some conditions for A,B and W. Here S^d,W denotes the generalized W-weighted Drazin inverse of a linear bounded operator S on a Banach space.     

Pairs of mutually annihilating operators

Pairs (A,B) of mutually annihilating operators AB=BA=0 on a finite dimensional vector space over an algebraically closed field were classified by Gelfand and Ponomarev [Russian Math. Surveys 23 (1968) 1-58] by method of linear relations. The classification of (A,B) over any field was derived by Nazarova, Roiter, Sergeichuk, and Bondarenko [J. Soviet Math. 3 (1975) 636-654] from the classification of finitely generated modules over a dyad of two local Dedekind rings. We give canonical matrices of (A,B) over any field in an explicit form and our proof is constructive: the matrices of (A,B) are sequentially reduced to their canonical form by similarity transformations (A,B)?(S^-1AS,S^-1BS).     

On Supersets of Wavelet Sets

Considering a single dyadic orthonormal wavelet ψ in L ²(?), it is still an open problem whether the support of $\widehat{\psi}$ always contains a wavelet set. As far as we know, the only result in this direction is that if the Fourier support of a wavelet function is “small” then it is either a wavelet set or a union of two wavelet sets. Without assuming that a set S is the Fourier support of a wavelet, we obtain some necessary conditions and some sufficient conditions for a “small” set S to contain a wavelet set. The main results, which are in terms of the relationship between two explicitly constructed subsets A and B of S and two subsets T ₂ and D ₂ of S intersecting itself exactly twice translationally and dilationally respectively, are (1) if $A\cup B\not\subseteq T_{2}\cap D_{2}$ then S does not contain a wavelet set; and (2) if A∪B?T ₂∩D ₂ then every wavelet subset of S must be in S?(A∪B) and if S?(A∪B) satisfies a “weak” condition then there exists a wavelet subset of S?(A∪B). In particular, if the set S?(A∪B) is of the right size then it must be a wavelet set.     

Goldie Ranks of Skew Power Series Rings of Automorphic Type

Let A be a semprime, right noetherian ring equipped with an automorphism α, and let B: = A[[y; α]] denote the corresponding skew power series ring (which is also semiprime and right noetherian). We prove that the Goldie ranks of A and B are equal. We also record applications to induced ideals.