Commutative Ideals Research Articles

Multiplication and mixed differentiation in generalized Fock spaces

The classical distribution theory founded by Laurent Schwartz [26] has an intrinsic multiplication problem. He indicates this in his paper [27]. This problem is clearly demonstrated by selecting two particular distributions and attempting to define a multiplication that enjoys commutative, associative, and unit identity properties. This process leads to the contradiction, 1 = eP I [lo]. A new generalized function theory developed by J. Colombeau [4-91 provides a solution to the multiplication problem and also develops a general integration theory. T. Todorov reexamines the new generalized function theory in the spirit of non-denumerable sequences termed ultrapowers [29]. The L. Schwartz distribution theory was reexamined in the spirit of fundamental sequences [ 16,333. However, there are new considerations in the theory of new generalized function theory requiring an algebraic notion of a particular ideal contained in a specified test function space. A brief review of these constructions can be found in Section 7 and a comprehensive review can be found in Ref. [25]. A principle application for our development will be modeling a physical system consisting of an “infinite” number of particles. A system of n-particles termed bosoms or fermions has previously been modeled by n-dimensional symmetric or antisymmetric functions. When this system is enlarged to an infinite number of particles, the model for the state space vector has the form,

Commutator ideals and semicommutator ideals of Toeplitz algebras associated with flows

We prove that for a Toeplitz C ∗ {C^ * } -algebra associated with a flow that has no fixed points, the commutator ideal and the semicommutator ideal coincide.

Remark on commutative approximate identities on homogeneous groups

We give, using the functional calculus of Hulanicki [4], a construction of a commutative approximate identity on every homogeneous group.

Smoothing properties and existence of solutions for the generalized Benjamin-Ono equation

We study the Cauchy problem for the generalized Benjamin-Ono equation ∂ t u − D | D| 2 μ u = DV'( u), (∗) where D = d dx , U > 0 , and V ϵ E 1( R,R ) with V(0) = V′(0) = 0. Using commutator identities and estimates, we prove that the equation (∗) exhibits smoothing properties similar to those of the generalized Korteweg-de Vries equation (GKdV), which is the special case μ = 1 of (∗). If V satisfies suitable estimates at zero and at infinity, we prove that the Cauchy problem for the equation (∗) with initial data u(0) = u 0 has a solution u ϵ L ∞( R , l 2)∪ L 2 loc( R , H μ loc) if u 0 ϵ L 2, and has a solution u ϵ L ∞( R , H μ)∪ L 2 loc( R , H 2μ loc)if u 0 ϵ H μ (finite energy solution). In addition, we prove that the usual Benjamin-Ono equation ( μ = 1 2 , V′(u) = u 2 ) has a solution u ϵ L ∞( R , H 1)∪ L 2 loc( R , H 3 2 loc) for u 0 ϵ H 1. Those results are easily extended to related equations such as the intermediate long wave equation or the Smith equation.

The Baker–Campbell–Hausdorff formula and nested commutator identities

The coefficients of the Baker–Campbell–Hausdorff expansion are calculated by using various methods. Comparison of the results yields several remarkable identities satisfied by multiple commutators, which, in turn, allow us to greatly simplify the form of the expansion up to order eight.

Commutator relations and identities in lattice-ordered groups.

Commutator relations and identities in lattice-ordered groups.

Remarks on a remark of Kaplansky

In his book

On the classification of commutator ideals

We compute the K-theory of the commutator ideals associated with almost periodic Toeplitz operators and use the K 0-group to classify these ideals.

Stable semi-groups of measures as commutative approximate identities on non-graded homogeneous groups

Stable semi-groups of measures as commutative approximate identities on non-graded homogeneous groups

Tarski's problem for varieties of groups with a commutator identity

AbstractIt is proved that for a variety of groups in which the relatively free groups are solvable, the relatively free groups of distinct finite rank are not elementarily equivalent.

Local Heisenberg inequalities

Heisenberg-like inequalities are derived from commutator identities, linking the quadratic dispersion in momentum to several functions of the position variable (or conversely). They have various physical and mathematical applications, and provide new links between the quadratic widths and the overall and mean peak widths of wave-functions.

Explicit expressions for Bezoutians

The following commutator identity is proved: [u(S ∗), v(S)] = [v 1(S ∗), u 1(S)] . Here S is the n by n matrix of the truncated shift operator S = ( Γ i , i+1), i = 0, 1,…, n − 1, and u, v are two polynomials of degree not exceeding n. The reciprocal polynomial f; 1 of a polynomial f; of degree ⩽ n is defined by f 1(z) = z nf( 1 z ) . The commutator identity is closely related to some properties of the Bezoutian matrix of a pair of polynomials; it is used to obtain the Bezoutian matrix in the form of a simple expression in terms of S and S ∗. To demonstrate the advantage of this expression, we show how it can be used to obtain simple proofs of some interesting corollaries.

On commutative approximate identities on non-graded homogeneous groups

On commutative approximate identities on non-graded homogeneous groups

Nonnumeric Computer Applications to Algebra, Trigonometry and Calculus

Algebra. Early automatic simplification programs employed an assemblage of ad hoc techniques such as collecting similar terms or factors, and employing transformations such as for all u: 1 U U, Ou*0, 0+ U--u. This approach is adequate for elementary simplification. However, the more sophisticated operations such as cancellation of polynomial greatest common divisors, factorization, solution of nonlinear equations, and simplification of radicals are far more efficient and easier to implement using a more systematic algebraic approach. Diverse topics make important contributions here. Particularly relevant topics are ring theory (including commutative, noncommutative and principal ideals), integral domains, Euclidean domains, unique factorization domains, field theory (including Galois, finite and quotient fields), algebraic geometry (including singular perturbations and catastrophe theory), p-adic numbers and valuation theory. Conversely, computer algebra can serve as a powerful tool for abstract algebra: Some systems have facilities that make it particularly easy to implement new algebraic structures, thus facilitating an empirical (perhaps exhaustive) study of some of their properties. For example, MACSYMA permits one to establish a new operator and declare that it has certain properties such as commutativity, associativity, left distributivity, etc. MACSYMA also has a so-called pattern matcher that makes it easy to establish automatic substitution rules such as q -k and k2 2_ 1. With the help of these mechanisms it is particularly easy to implement an additional known algebraic structure (such as quaternions) or to implement an original algebraic structure in order to help study its properties. MACSYMA also has a supplementary package that provides particularly strong support for investigations of commutative rings. For example, to establish R as the ring of polynomials in x, y, z and w over the integers mod 231 1, mod Z2 _ W3, one merely enters the commands

Narrowness implies uniformity

This paper is about varietiesV of universal algebras which satisfy the following numerical condition on the spectrum: there are only finitely many prime integersp such thatp is a divisor of the cardinality of some finite algebra inV. Such varieties are callednarrow. The variety (or equational class) generated by a classK of similar algebras is denoted by V(K)=HSPK. We define Pr (K) as the set of prime integers which divide the cardinality of a (some) finite member ofK. We callK narrow if Pr (K) is finite. The key result proved here states that for any finite setK of finite algebras of the same type, the following are equivalent: (1) SPK is a narrow class. (2) V(K) has uniform congruence relations. (3) SK has uniform congruences and (3) SK has permuting congruences. (4) Pr (V(K))= Pr(SK). A varietyV is calleddirectly representable if there is a finite setK of finite algebras such thatV= V(K) and such that all finite algebras inV belong to PK. An equivalent definition states thatV is finitely generated and, up to isomorphism,V has only finitely many finite directly indecomposable algebras. Directly representable varieties are narrow and hence congruence modular. The machinery of modular commutators is applied in this paper to derive the following results for any directly representable varietyV. Each finite, directly indecomposable algebra inV is either simple or abelian.V satisfies the commutator identity [x,y]=x·y·[1,1] holding for congruencesx andy over any member ofV. The problem of characterizing finite algebras which generate directly representable varieties is reduced to a problem of ring theory on which there exists an extensive literature: to characterize those finite ringsR with identity element for which the variety of all unitary leftR-modules is directly representable. (In the terminology of [7], the condition is thatR has finite representation type.) We show that the directly representable varieties of groups are precisely the finitely generated abelian varieties, and that a finite, subdirectly irreducible, ring generates a directly representable variety iff the ring is a field or a zero ring.

A characterization of finite p-soluble groups of p-length one by commutator identities

AbstractA classical result of M. Zorn states that a finite group is nilpotent if and only if it satisfies an Engel condition. If this is the case, it satisfies almost all Engel conditions. We shall give a similar description of the class of p-soluble groups of p-length one by a sequence of commutator identities.

Remarks concerning sums of three squares and quaternion commutator identities

A survey of identities connected with sums of three squares, 2-dimensional subalgebras of the quaternions, and additive commutators is given. The study of the last is linked to a theorem already observed by Gauss and re-proved here in various ways. A study of generalizations in recent years is added.

A commutator identity for basic p-groups

AbstractThe consequences of the law (Ga(1), hellip, Ga(n)) = 1 in a basic p-group are examined. The principal tools are a combinatorial analysis of the lattice of n-tuples of positive integers and a theorem of the author about higher-commutator subgroups.Subject classification (Amer. Math. Soc. (MOS) 1970): 20 D 15.

Inertial coefficient rings and the idempotent lifting property

A commutative ring R R with identity is called an inertial coefficient ring if every finitely generated R R -algebra A A with A / N A/N separable over R R contains a separable R R -subalgebra S S of A A such that A = S + N A = S + N , where N N is the Jacobson radical of A A . We say A A has the idempotent lifting property if every idempotent in A / N A/N is the image of an idempotent in A A . Our main theorem is that any finitely generated algebra over an inertial coefficient ring has the idempotent lifting property.

Applications of graph theory to matrix theory

Let A 1 , … , A k {A_1}, \ldots ,{A_k} be n × n n \times n matrices over a commutative ring R R with identity. Graph theoretic methods are established to compute the standard polynomial [ A 1 , … , A k ] [{A_1}, \ldots ,{A_k}] . It is proved that if k > 2 n − 2 k > 2n - 2 , and if the characteristic of R R either is zero or does not divide 4 I ( 1 / 2 n ) − 2 4I(1/2n) - 2 , where I I denotes the greatest integer function, then there exist n × n n \times n skew-symmetric matrices A 1 , … , A k {A_1}, \ldots ,{A_k} such that [ A 1 , … , A k ] ≠ 0 [{A_1}, \ldots ,{A_k}] \ne 0 .