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We study commutative algebras satisfying the identity [Formula: see text] It is known that for [Formula: see text] and for characteristic not [Formula: see text] or [Formula: see text], the algebra is a commutative power-associative algebra. These algebras have been widely studied by Albert, Gerstenhaber and Schafer. For [Formula: see text] Guzzo and Behn in 2014 proved that commutative algebras of dimension [Formula: see text] satisfying [Formula: see text] are solvable. We consider the remaining values of [Formula: see text] We prove that commutative algebras satisfying [Formula: see text] with [Formula: see text] and generated by one element are nilpotent of nilindex [Formula: see text] (we assume characteristic of the field [Formula: see text]).

In this article, we study third power associative, antiflexible rings satisfying the identity (a,b,bc)=b(a,b,c). We prove that third power associative, antiflexible rings satisfying the identity (a,b,bc)=b(a,b,c) with characteristic ≠2,3 are associative of degree five. As a consequence of this result, we prove that a third power associative semiprime antiflexible ring satisfying the identity (a,b,bc)=b(a,b,c) is associative.

When a commutative nilalgebra has nilidex four, we can separate their study in three big cases. One of them is when the algebra \({\mathcal {A}}\) satisfies the identity \( (((yx)x)x)x= 0\). In this paper we get results on the structure of this kind of algebras. Let W be the subspace of \({\mathcal {A}}\) generated by the cubes. We prove that \(W^2 = 0\) and we use this to get others properties of W.

This paper deals with two varieties of commutative non-associative algebras. One variety satisfies L x 3 + L x 3 = 0 . The other variety satisfies L x 3 = 0 . We prove that in either variety, any finitely generated algebra is nilpotent. Our results require characteristic ≠ 2 , 3 .

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 − ω ( x ) x ) 2 = 0 is satisfied. We also find sufficient conditions for A to have idempotents.

Albert's construction for commutative semifields of order 2 n , n odd, is presented. It avoids the construction of a presemifield and, in the case that n is prime, allows us to determine automorphism groups and the isomorphism classes. If n is a prime greater than three, the semifields are strictly not associative. These semifields are new for all n greater than three, differing from the binary semifields in that each admits only the trivial automorphism. The authors present an explicit construction of an isotope of the 25-element semifield that contains a subsemifield of order 22.

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 and time. The calculations can be done in characteristic zero or any prime greater than 6, and similar results are expected. We generated the nuclear elements using prime 103. We check our answer using five other primes.

A finite semifield D is a finite nonassociative ring with identity such that the set D* = D \{0} is a loop under the product. Wene conjectured in [1] that any finite semifield is either right or left primitive, i.e. D* is the set of right (or left) principal powers of an element in D. In this paper we study the primitivity of finite semifields with 64 and 81 elements.

A sign pattern Z (a matrix whose entries are elements of {+,−,0}) is spectrally arbitrary if for any self-conjugate spectrum there is a real matrix with sign pattern Z having the given spectrum. Spectrally arbitrary sign patterns were introduced in [J.H. Drew, C.R. Johnson, D.D. Olesky, P. van den Driessche, Spectrally arbitrary patterns, Linear Algebra Appl. 308 (2000) 121–137], where it was (incorrectly) stated that if a sign pattern Z is reducible and each of its irreducible components is a spectrally arbitrary sign pattern, then Z is a spectrally arbitrary sign pattern, and it was conjectured that the converse is true as well; we present counterexamples to both of these statements. In [T. Britz, J.J. McDonald, D.D. Olesky, P. van den Driessche, Minimal spectrally arbitrary patterns, SIAM J. Matrix Anal. Appl. 26 (2004) 257–271] it was conjectured that any n×n spectrally arbitrary sign pattern must have at least 2n nonzero entries; we establish that this conjecture is true for 5×5 sign patterns. We also establish analogous results for nonzero patterns.

The set of real matrices described by a sign pattern (a matrix whose entries are elements of {+, −, 0}) has been studied extensively but only loose bounds were available for the minimum rank of a tree sign pattern. A simple graph has been associated with the set of symmetric matrices having a zero–nonzero pattern of off-diagonal entries described by the graph, and the minimum rank/maximum eigenvalue multiplicity among matrices in this set is readily computable for a tree. In this paper, we extend techniques for trees to tree sign patterns and trees allowing loops (with the presence or absence of loops describing the zero–nonzero pattern of the diagonal), allowing precise computation of the minimum rank of a tree sign pattern and a tree allowing loops. For a symmetric tree sign pattern or a tree that allows loops, we provide an algorithm that allows exact computation of maximum multiplicity and minimum rank, and can be used to obtain a symmetric integer matrix realizing minimum rank.

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.

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.

We shall study representations of algebras over fields of characteristic ≠2, 3 of dimension 4 which satisfy the identities xy − yx = 0, and (( xx ) x ) x = 0. In these algebras the multiplication operator was shown to be nilpotent by [I. Correa, R. Hentzel, A. Labra, On the nilpotence of the multiplication operator in commutative right nilalgebras, Commun. Alg. 30 (7) (2002) 3473–3488]. In this paper we use this result in order to prove that there are no non-trivial one-dimensional representations, there are only reducible two-dimensional representations, and there are irreducible and reducible three-dimensional representations.

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.

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...

We study conditions under which the identity in a commutative nonassociative algebra A implies is nilpotent where is the multiplication operator for all in A. The separate conditions that we found to be sufficient are (1) dimension four or less, (2) any additional non-trivial identity of degree four, or (3) We assume characteristic

The Cayley-Dickson process gives a recursive method of constructing a nonassociative algebra of dimension 2 n for all n ≥ 0, beginning with any ring of scalars. The algebras in this sequence are known to be flexible quadratic algebras; it follows that they are noncommutative Jordan algebras: they satisfy the flexible identity in degree 3 and the Jordan identity in degree 4. For the integral sedenion algebra (the double of the octonions) we determine a complete set of generators for the multilinear identities in degrees ≤ 5. Since these identities are satisfied by all flexible quadratic algebras, it follows that a multilinear identity of degree ≤ 5 is satisfied by all the algebras obtained from the Cayley-Dickson process if and only if it is satisfied by the sedenions.

We show that noncommutative power-associative nilalgebras of finite dimension n and nilindex k are solvable if k = n + 1 or k = n . For any given integer n > 2, we present an example of a power-associative nilalgebra of dimension n and nilindex n − 1 which is not solvable. This implies a power-associative nilalgebra of dimension n and nilindex k need not be solvable if k n .

We determine the identities of degree ≤ 9 satisfied by the new ternary operations abc + bca + cab + acb + bac + cba (symmetric sum), abc + bca + cab − acb − bac − cba (alternating sum), and abc + bca + cab (cyclic sum) on every triple system satisfying the total associativity identities ( abc ) de ≡ a ( bcd ) e ≡ ab ( cde ). These new operations are ternary analogs of the Lie and Jordan products. The identities they satisfy may be regarded as ternary versions of the Jacobi and Jordan identities.

Suppose that R = ( R, +, ) is a nonassociative ring, and that R + = ( R, +, ˆ) is a Jordan ring where the ˆ-product is defined by x ˆ y = xy + yx. We compute the minimal degree three and degree four identities which are strong enough to make R third-power associative, R + Jordan, and the McCrimmon radical of R + to be an ideal of R . We display an absorption identity of R which demonstrates that the McCrimmon radical of R + is an ideal of R . Because of the sizes of the matrices involved, the work was done modulo various primes and extrapolated to the rational numbers. The goal was to present the absorption identities. Actual proofs are too large to publish. The interested reader may check them using computer algebra packages. For ( R + , +, ˆ) to have a McCrimmon radical, we have to have characteristic ≠ 2.The heart of the paper is a solution to the matrix equation AX = B where both the matrix A and the vector B have entries which are rational functions in α. The solution proceeds by solving the system modulo various primes for various values of α and using these residues to estimate the vector of rational functions X. A genetic algorithm is used to maximize the number of zero entries in the solution X.

We work in the space R n×n of n-by-n real matrices. We say that a linear transformation τ on this space is Toda-like if it maps symmetric matrices to skew-symmetric matrices. With such a transformation we associate a bilinear operation α defined by α(X, Y) ≔ [X, τY] + [Y, τX] where [U, V] ≔ UV − VU. Then R n×n together with this operation is a (usually not associative) algebra. We call any subalgebra of such an algebra a Toda-like algebra. We classify these algebras, which arise in connection with eigenvalue computations. We determine the Toda-like algebras which are of primary interest for this application. We accomplish this classification by studying certain difference-weighted graphs which we associate with the algebras.

We compute the identities and the central identities of degree ≤ 6 of the Cayley–Dickson algebras. Our process can be used to compute identities of higher degree. We assume a field of characteristic 0 or greater than the degree of the identities studied. Our identities are homogeneous multilinear polynomials.

Let $R$ be a nonassociative ring, and $N = (R,R,R) + [R,(R,R,R)]$. We show that $W = \{w\in N | Rw +wR +R(wR) \subset N\}$ is a two-sided ideal of $R$. If for some $r\in R$, any one of the sets $(r, R, R)$, $(R,r, R)$ or $(R, R,r)$ is contained in $W$, then the other two sets are contained in $W$ also. If the associators are assumed to be contained in either the left, the middle, or the right nucleus, and $I$ is the ideal generated by all associators, then $I^2 \subset W$. If $N$ is assumed to be contained in the left or the right nucleus, then $W^2 = 0$. We conclude that if $R$ is semiprime and $N$ is contained in the left or the right nucleus, then $R$ is associative. We assume characteristic not 2.

Let $R$ be a nonassociative ring, and $N$ the left nucleus. It is shown that if $R$ is a simple ring satisfying $(R,R,R) \subset N$ and $((R,R,R),R) \subset N$ and char $R\neq 2$, then $R$ is associative.

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 asociative. This generalizes a recent result by Y. Paul [7].

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.

In this paper we study flexible algebras (possibly infinite-dimensional) satisfying the polynomial identity x(yz) = y(zx). We prove that in these algebras, products of five elements are associative and commutative. As a consequence of this, we get that when such an algebra is a nil-algebra of bounded nil-index, it is nilpotent. Furthermore, we obtain optimal bounds for the index of nilpotency. Another consequence that we get is that these algebras are associative when they are semiprime.

Given ann-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).

We give a simple characterization for a nonassociative algebra [Formula: see text], having characteristic ≠2, to be commutative. Namely, [Formula: see text] is commutative if and only if it is flexible with a commuting set of generators. A counterexample shows that characteristic ≠2 is necessary. Both the characterization and the counterexample were discovered using the computer algebra system in [2].

Let ( A , ω) be a finite dimensional Bernstein algebra and N the kernel of ω. We study the algebras where dim N 2 is 1. The algebras fall into two general classes. For the first of these classes we give the multiplication tables for the complete set of nonisomorphic algebras. For the second of these classes we give the multiplication tables for what we call “complete algebras.” We show that any algebra of the second class can be embedded in a complete algebra. The multiplication in the complete algebras is easy to describe. The Bernstein algebras of the second class are then characterized as subalgebras of the complete algebras. For Jordan Bernstein algebras satisfying dim N 2 = 1 we give the complete classification.

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 demonstr...

We extend the concept of commutative nilalgebras to commutative algebras which are not power associative. We shall study commutative algebras A over fields of characteristic ≠ 2, 3 which satisfy the identities x(x(xx)) = 0 and β{x(y(xx)) - x(x(xy))} + γ{y(x(xx)) - x(x(xy))} = 0. In these algebras the multiplication operator was shown to be nilpotent by Correa, Hentzel and Labra [2]. In this paper we prove that for every x ∈ A we have A(A((xx)(xx))) = 0. We prove that there is an ideal I of A satisfying AI = IA = 0 and A/I is power associative.

Let F be an algebraically closed field of characteristic zero and L an RA loop. We prove that the loop algebra FL is in the variety generated by the split Cayley–Dickson algebra Z F over F. For RA2 loops of type M(Dih(A), −1,g 0), we prove that the loop algebra is in the variety generated by the algebra  3 which is a noncommutative simple component of the loop algebra of a certain RA2 loop of order 16. The same does not hold for the RA2 loops of type M(G, −1,g 0), where G is a non-Abelian group of exponent 4 having exactly 2 squares.

The free nonassociative algebra has two subspaces which are closed under both the commutator and the associator: the Akivis elements and the primitive elements. Every Akivis element is primitive, but there are primitive elements which are not Akivis. Using a theorem of Shestakov, we give a recursive formula for the dimension of the Akivis elements. Using a theorem of Shestakov and Umirbaev, we prove a closed formula for the dimension of the primitive elements. These results generalize the Witt dimension formula for the Lie elements in the free associative algebra.

We prove that commutative power-associative nilalgebras of dimension 6 over a field of characteristic ≠2,3,5 are solvable.

We consider some polynomial identities of degree ≤ 5 which are satisfied by all symmetric quadratic algebras. We call rings satisfying these identities generalized quadratic rings , or GQ-rings . We show that when the ring is not flexible, these identities are enough to make the ring quadratic over its center. Therefore, simple nonflexible GQ-rings are symmetric quadratic algebras over their center, which is a field. For prime GQ-rings, the center has no nonzero zero divisors. Prime GQ-rings, which are not flexible, are subrings of the quadratic algebra formed by extending the center to its field of quotients. Flexible GQ-rings are noncommutative Jordan rings which satisfy a quadratic condition over their commutative center. We show that any semi-prime GQ-ring is a subdirect sum of a noncommutative Jordan ring (which satisfies a quadratic condition over its commutative center) and a nonflexible ring (which is symmetric quadratic over its center).

We use a computer to verify that the ideal N of all weight zero elements of any (not necessarily finite dimensional) Bernstein algebra is solvable of index ≤4. We also use a computer to verify that N 2 is nilpotent of index ≤9. We give three examples of Bernstein algebras which show that various hypotheses like finite dimensionality, finitely generatedA 2 = A, are separately not enough to force N to be nilpotent.