Some results on the structure of a class of commutative power-associative nilalgebras of nilindex four (original) (raw)

On a Class of Commutative Power-Associative Nilalgebras

Journal of Algebra, 1999

We prove that commutative power associative nilalgebras of nilindex n and dimension n are nilpotent of index n. We find a necessary and sufficient condition for such an algebra to be a Jordan algebra and give all corresponding isomorphism classes.

Solvability of Commutative Power-Associative Nilalgebras of Nilindex 4 and Dimension

Proyecciones (Antofagasta), 2004

Let A be a commutative power-associative nilalgebra. In this paper we prove that when A (of characteristic 6 = 2) is of dimension ≤ 8 and x 4 = 0 for all x ∈ A, then ((A 2) 2) 2 = 0. That is, A is solvable. We conclude that if A is of dimension ≤ 7 over a field of characteristic 6 = 2, 3 and 5, then A is solvable.

Nilpotence of a class of commutative power-associative nilalgebras

Journal of Algebra, 2005

Let A be a commutative algebra over a field F of characteristic = 2, 3. In [M. Gerstenhaber, On nilalgebras and linear varieties of nilpotent matrices II, Duke Math. J. 27 (1960) 21-31], M. Gerstenhaber proved that if A is a nilalgebra of bounded index t and the characteristic of F is zero (or greater than 2t − 3), then the right multiplication R x

Commutative Jordan Nilalgebras of Nilindex 4 Generated by at Most Three Elements

JP journal of algebra, number theory and applications, 2022

The present study is about commutative Jordan nilalgebras of nilindex 4 generated by at most three elements over a field K of characteristic. 3 , 2 ≠ We first look at the case where such algebras are generated by two elements and satisfy Engel's third or fourth condition, then we consider the case where they are generated by three elements and satisfy Engel's third condition. We prove that these algebras are nilpotent, and hence solvable, while giving a bound of the index of solvability and nilpotency.

Nilpotent linear transformations and the solvability of power-associative nilalgebras

Linear Algebra and Its Applications, 2005

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.

On the Classification of Commutative Right-Nilalgebras of Dimension at Most Four

Communications in Algebra, 2007

Gerstenhaber and Myung in [10] classified all commutative-power associative nilalgebras of dimension 4. In [7] Gerstenhaber and Myung's results are generalized by giving a classification of commutative right-nilalgebras of right-nilindex four and dimension at most four, without assuming power-associativity. In this paper we complete this research and give a classification of commutative right-nilalgebras of right-nilindex five and dimension four, without assuming power-associativity, thus completing the classification of commutative right-nilalgebras of dimension at most four.

On Solvability of Noncommutative Power-Associative Nilalgebras

Journal of Algebra, 2001

We show that noncommutative power-associative nilalgebras of finite dimension n and nilindex k are solvable if k s n q 1 or k s n. For any given integer n ) 2, we present an example of a power-associative nilalgebra of dimension n and nilindex n y 1 which is not solvable. This implies a power-associative nilalgebra of dimension n and nilindex k need not be solvable if kn. ᮊ 2001 Academic Press Recall that a nonassociative algebra J is called sol¨able if the descend-Ž1.