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Articles 1 - 22 of 22
Full-Text Articles in Algebra
Experimenting With The Identity (Xy)Z = Y(Zx), Irvin Roy Hentzel, David P. Jacobs, Sekhar V. Muddana
Experimenting With The Identity (Xy)Z = Y(Zx), Irvin Roy Hentzel, David P. Jacobs, Sekhar V. Muddana
Irvin Roy Hentzel
An experiment with the nonassociative algebra program Albert led to the discovery of the following surprising theorem. Let G be a groupoid satisfying the identity (xy)z = y(zx). Then for products in G involving at least five elements, all factors commute and associate. A corollary is that any semiprime ring satisfying this identity must be commutative and associative, generalizing a known result of Chen.
Semiprime Locally(-1, 1) Ring With Minimal Condition, Irvin R. Hentzel, H. F. Smith
Semiprime Locally(-1, 1) Ring With Minimal Condition, Irvin R. Hentzel, H. F. Smith
Irvin Roy Hentzel
Let L be a left ideal of a right alternative ring A with characteristic ::/=2. If L is maximal and nil, then L is a two-sided ideal. If L is minimal, then it is either a two-sided ideal, or the ideal it generates is contained in the right nucleus of A. In particular, if A is prime, then a minimal left ideal of A must be a two-sided ideal. Let A be a semiprime locally (-1, 1) ring with characteristic ::1=2, 3. Then A is isomorphic to a subdirect sum of an alternative ring, a strong (-1, 1) ring, and …
Minimal Identities Of Bernstein Alegebras, Irvin R. Hentzel, Ivan Correa, Luiz Antionio Peresi
Minimal Identities Of Bernstein Alegebras, Irvin R. Hentzel, Ivan Correa, Luiz Antionio Peresi
Irvin Roy Hentzel
We construct the minimal identities for Bernstein algebras, exceptional Bernstein algebras and normal Bernstein algebras. We use the technique of processing identities via the representation of the symmetric groups. The computer algorithms for creating the standard tableaus and the integral representations are summarized.
Counterexamples In Nonassociative Algebra, Irvin R. Hentzel, Luiz Antonio Peresi
Counterexamples In Nonassociative Algebra, Irvin R. Hentzel, Luiz Antonio Peresi
Irvin Roy Hentzel
We present a method of constructing counterexamples in nonassociative algebra. The heart of the computation is constructing a matrix of identities and reducing this matrix (usually very sparse) to row canonical form. The example is constructed from the entries in one column of this row canonical form. While this procedure is not polynomial in the degree of the identity, several shortcuts are listed which shorten calculations. Several examples are given.
On Preserving Structured Matrices Using Double Bracket Operators: Tridiagonal And Toeplitz Matrices, Kenneth Driessel, Irvin R. Hentzel, Wasin So
On Preserving Structured Matrices Using Double Bracket Operators: Tridiagonal And Toeplitz Matrices, Kenneth Driessel, Irvin R. Hentzel, Wasin So
Irvin Roy Hentzel
In the algebra of square matrices over the complex numbers, denotes Two problems are solved: (1) Find all Hermitian matrices M which have the following property: For every Hermitian matrix A, if A is tridiagonal, then so is (2) Find all Hermitian matrices M which have the following property: For every Hermitian matrix A, if A is Toeplitz, then so is
On Prime Right Alternative Algebras And Alternators, Giulia Maria Piacentini Cattaneo, Irvin R. Hentzel
On Prime Right Alternative Algebras And Alternators, Giulia Maria Piacentini Cattaneo, Irvin R. Hentzel
Irvin Roy Hentzel
We study subvarieties of the variety of right alternative algebras over a field of characteristic t2,t3 such that the defining identities of the variety force the span of the alternators to be an ideal and do not force an algebra with identity element to be alternative. We call a member of such a variety a right alternative alternator ideal algebra. We characterize the algebras of this subvariety by finding an identity which holds if and only if an algebra belongs to the subvariety. We use this identity to prove that if R is a prime, right alternative alternator ideal algebra …
Nilpotent Linear Transformations And The Solvability Of Power-Associative Nilalgebras, Ivan Correa, Irvin R. Hentzel, Pedro Pablo Julca, Luiz Antonio Peresi
Nilpotent Linear Transformations And The Solvability Of Power-Associative Nilalgebras, Ivan Correa, Irvin R. Hentzel, Pedro Pablo Julca, Luiz Antonio Peresi
Irvin Roy Hentzel
We prove some results about nilpotent linear transformations. As an application we solve some cases of Albert’s problem on the solvability of nilalgebras. More precisely, we prove the following results: commutative power-associative nilalgebras of dimension n and nilindex n − 1 or n − 2 are solvable; commutative power-associative nilalgebras of dimension 7 are solvable.
Nuclear Elements Of Degree 6 In The Free Alternative Algebra, Irvin R. Hentzel, L. A. Peresi
Nuclear Elements Of Degree 6 In The Free Alternative Algebra, Irvin R. Hentzel, L. A. Peresi
Irvin Roy Hentzel
We construct five new elements of degree 6 in the nucleus of the free alternative algebra. We use the representation theory of the symmetric group to locate the elements. We use the computer algebra system ALBERT and an extension of ALBERT to express the elements in compact form and to show that these new elements are not a consequence of the known degree-5 elements in the nucleus. We prove that these five new elements and four known elements form a basis for the subspace of nuclear elements of degree 6. Our calculations are done using modular arithmetic to save memory …
Solvability Of Commutative Right-Nilalgebras Satisfying (B (Aa)) A= B ((Aa) A), Ivan Correa, Alicia Labra, Irvin R. Hentzel
Solvability Of Commutative Right-Nilalgebras Satisfying (B (Aa)) A= B ((Aa) A), Ivan Correa, Alicia Labra, Irvin R. Hentzel
Irvin Roy Hentzel
We study commutative right-nilalgebras of right-nilindex four satisfying the identity (b(aa))a = b((aa)a). Our main result is that these algebras are solvable and not necessarily nilpotent. Our results require characteristic ≠ 2, 3, 5.
Invariant Nonassociative Algebra Structures On Irreducible Representations Of Simple Lie Algebras, Murray Bremner, Irvin R. Hentzel
Invariant Nonassociative Algebra Structures On Irreducible Representations Of Simple Lie Algebras, Murray Bremner, Irvin R. Hentzel
Irvin Roy Hentzel
An irreducible representation of a simple Lie algebra can be a direct summand of its own tensor square. In this case, the representation admits a nonassociative algebra structure which is invariant in the sense that the Lie algebra acts as derivations. We study this situation for the Lie algebra sl(2).
The Nucleus Of The Free Alternative Algebra, Irvin R. Hentzel, L. A. Peresi
The Nucleus Of The Free Alternative Algebra, Irvin R. Hentzel, L. A. Peresi
Irvin Roy Hentzel
We use a computer procedure to determine a basis of the elements of degree 5 in the nucleus of the free alternative algebra. In order to save computer memory, we do our calculations over the field Z103. All calculations are made with multilinear identities. Our procedure is also valid for other characteristics and for determining nuclear elements of higher degree.
Rings With (A, B, C) = (A, C, B) And (A, [B, C]D) = 0: A Case Study Using Albert, Irvin R. Hentzel, D. P. Jacobs, Erwin Kleinfeld
Rings With (A, B, C) = (A, C, B) And (A, [B, C]D) = 0: A Case Study Using Albert, Irvin R. Hentzel, D. P. Jacobs, Erwin Kleinfeld
Irvin Roy Hentzel
Albert is an interactive computer system for building nonassociative algebras [2]. In this paper, we suggest certain techniques for using Albert that allow one to posit and test hypotheses effectively. This process provides a fast way to achieve new results, and interacts nicely with traditional methods. We demonstrate the methodology by proving that any semiprime ring, having characteristic ≠ 2, 3, and satisfying the identities (a, b, c) - (a, c, b) = (a, [b, c], d) = 0, is associative. This generalizes a recent result by Y. Paul [7].
A Variety Containing Jordan And Pseudo-Composition Algebras, Irvin R. Hentzel, Luiz Antonio Peresi
A Variety Containing Jordan And Pseudo-Composition Algebras, Irvin R. Hentzel, Luiz Antonio Peresi
Irvin Roy Hentzel
We consider 3-Jordan algebras, i.e., the nonassociative commutative algebras satisfying (x^3 y)x=x^3(yx). The variety of 3-Jordan algebras contains all Jordan algebras and all pseudo-composition algebras. We prove that a simple 3-Jordan algebra with idempotent is either a Jordan algebra or a pseudo-composition algebra.
Commutative Finitely Generated Algebras Satisfying ((Yx)X)X=0 Are Solvable, Ivan Correa, Irvin R. Hentzel
Commutative Finitely Generated Algebras Satisfying ((Yx)X)X=0 Are Solvable, Ivan Correa, Irvin R. Hentzel
Irvin Roy Hentzel
No abstract provided.
Identities Relating The Jordan Product And The Associator In The Free Nonassociative Algebra, Murray R. Bremner, Irvin R. Hentzel
Identities Relating The Jordan Product And The Associator In The Free Nonassociative Algebra, Murray R. Bremner, Irvin R. Hentzel
Irvin Roy Hentzel
We determine the identities of degree ≤ 6 satisfied by the symmetric (Jordan) product a○b = ab + ba and the associator [a,b,c] = (ab)c - a(bc) in every nonassociative algebra. In addition to the commutative identity a○b = b○a we obtain one new identity in degree 4 and another new identity in degree 5. We demonstrate the existence of further new identities in degree 6. These identities define a variety of binary-ternary algebras which generalizes the variety of Jordan algebras in the same way that Akivis algebras generalize Lie algebras.
Generalized Alternative And Malcev Algebras, Irvin R. Hentzel, H.F. Smith
Generalized Alternative And Malcev Algebras, Irvin R. Hentzel, H.F. Smith
Irvin Roy Hentzel
No abstract provided.
Generalized Right Alternative Rings, Irvin R. Hentzel
Generalized Right Alternative Rings, Irvin R. Hentzel
Irvin Roy Hentzel
We show that weakening the hypotheses of right alternative rings to the three identities (1) (ab,c,d) + (a,b,[c,d]) = a(b,c,d) + (a,c,d)b (2) (α,α,α) = 0 (3) ([a,b],b,b) = O for all α, b, c, d in the ring will not lead to any new simple rings. In fact, the ideal generated by each associator of the form (a, b, b) is a nilpotent ideal of index at most three. Our proofs require characteristic ^2 , ^3 .
Fast Change Of Basis In Algebras, Irvin R. Hentzel, David Pokrass Jacobs
Fast Change Of Basis In Algebras, Irvin R. Hentzel, David Pokrass Jacobs
Irvin Roy Hentzel
Given an n-dimensional algebraA represented by a basisB and structure constants, and given a transformation matrix for a new basisC., we wish to compute the structure constants forA relative to C. There is a straightforward way to solve this problem inO(n5) arithmetic operations. However given an O(nω) matrix multiplication algorithm, we show how to solve the problem in time O(nω+1). Using the method of Coppersmith and Winograd, this yields an algorithm ofO(n3.376).
Complexity And Unsolvability Properties Of Nilpotency, Irvin R. Hentzel, David Pokrass Jacobs
Complexity And Unsolvability Properties Of Nilpotency, Irvin R. Hentzel, David Pokrass Jacobs
Irvin Roy Hentzel
A nonassociative algebra is nilpotent if there is some n such that the product of any n elements, no matter how they are associated, is zero. Several related, but more general, notions are left nilpotency, solvability, local nilpotency, and nillity. First the complexity of several decision problems for these properties is examined. In finite-dimensional algebras over a finite field it is shown that solvability and nilpotency can be decided in polynomial time. Over Q, nilpotency can be decided in polynomial time, while the algorithm for testing solvability uses a polynomial number of arithmetic operations, but is not polynomial time. Also …
Idempotents In Plenary Train Algebras, Antonio Behn, Irvin R. Hentzel
Idempotents In Plenary Train Algebras, Antonio Behn, Irvin R. Hentzel
Irvin Roy Hentzel
In this paper we study plenary train algebras of arbitrary rank. We show that for most parameter choices of the train identity, the additional identity (x^2 -w(x)x)^2 =0 is satisfied. We also find sufficient conditions for A to have idempotents.
Left Centralizers On Rings That Are Not Semiprime, Irvin R. Hentzel, M.S. Tammam El-Sayiad
Left Centralizers On Rings That Are Not Semiprime, Irvin R. Hentzel, M.S. Tammam El-Sayiad
Irvin Roy Hentzel
A (left) centralizer for an associative ring R is an additive map satisfying T(xy) = T(x)y for all x, y in R. A (left) Jordan centralizer for an associative ring R is an additive map satisfying T(xy+yx) = T(x)y + T(y)x for all x, y in R. We characterize rings with a Jordan centralizer T. Such rings have a T invariant ideal I, T is a centralizer on R/I, and I is the union of an ascending chain of nilpotent …
Right Alternative Rings, Irvin Roy Hentzel