The Andrunakievich lemma for alternative rings

This article is published as Hentzel, Irvin Roy. “The Andrunakievich lemma for alternative rings,” Algebras, Groups, and Geometries 6 (1989), no. 1, 55-64. Posted with permission.</p

Generalized alternative and Malcev algebras

This article is published as Hentzel, I. R., and H. F. Smith. "Generalized alternative and Malcev algebras." Rocky Mountain Journal of Mathematics 12, no. 2 (1982): 367-374. doi: 10.1216/RMJ-1982-12-2-367. Posted with permission.</p

Generalized Right Alternative Rings

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

Identities relating the Jordan product and the associator in the free nonassociative algebra

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

Invariant Nonassociative Algebra Structures on Irreducible Representations of Simple Lie Algebras

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)

Counterexamples in Nonassociative Algebra

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

Complexity and Unsolvability Properties of Nilpotency

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 presented is a polynomial time probabilistic algorithm for deciding left nillity. Then a problem involving algebras given by generators and relations is considered and shown to be NP-complete. Finally, a relation between local left nilpotency and a set of natural numbers that is 1-complete for the class $\Pi_{2}$ in the arithmetic hierarchy of recursion theory is demonstrated

Semiprime locally (-1, 1) rings with minimal condition

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 a locally (-1, 1) ring with a certain property, where each of these three rings is likewise semiprime with characteristic ::/=2, 3, and where each satisfies any minimal condition satisfied by A. In particular, when A satisfies the minimal condition on two-sided ideals, then A satisfies certain additional identities; and when A satisfies the minimal condition on right ideals, then it must be alternative

Fast change of basis in algebras

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)

Commutative Finitely Generated Algebras Satisfying ((yx)x)x = 0 are Solvable

We study commutative, nonassociative algebras satisfying the identity (1) ((yx)x)x = 0 We show that finitely generated algebras over a field K of characteristic ≠ 2 satisfying (1) are solvable. For x in an algebra A, define the multiplicatin operator Rx by yRx = yx, for all y ∈ A. Our identify is then that Rx3 = 0