## About

29

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September 2004 - September 2015

## Publications

Publications (29)

A classical problem in nonassociative algebras involves the existence of simple finite-dimensional commutative nilalgebras. In this paper, we study the class Ω of nonassociative algebras satisfying the identity (xy)2=x2y2 over a field of characteristic different from 2 and 3. We show that every unitary algebra in Ω is associative. Next, we prove th...

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...

In this article, we prove the nilpotency of commutative nonassociative finitely generated algebras satisfying an identity of type (Formula presented.) with α + β ≠ 0. Our result requires characteristic ≠ 2, 3, 5.

This paper deals with two varieties of commutative non-associative algebras. One variety satisfies Lx3+Lx3=0. The other variety satisfies Lx3=0. We prove that in either variety, any finitely generated algebra is nilpotent. Our results require characteristic ≠2,3.

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 obta...

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 article we study nonassociative rings satisfying the polynomial identity x yz = y zx , which we call “cyclic rings.” We prove that every semiprime cyclic ring is associative and commutative and that every cyclic right-nilring is solvable. Moreover, we find sufficient conditions for the nilpotency of cyclic right-nilrings and apply these res...

We study conditions under which the identity ((xx)x)x=0 in a commutative nonassociative algebra A implies R x is nilpotent where R x is the multiplication operator R x (y)=xy for all y 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) ((xx)x)(x...

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 dim...

Using a factorization of quasi n-maps we find a relationship between the module formed by the n-maps and the module formed by the quasi n-maps. In particular, we characterize the quasi cubic forms using a relation called the parallelepiped law. Moreover we give necessary and su.cient conditions for the equality of the modules of quasi cubic forms a...

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

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 nee...

We prove that commutative power-associative nilalgebras of dimension 5 are solvable.

We prove that commutative power associative nilalgebras of nilindexnand dimensionnare nilpotent of indexn. We find a necessary and sufficient condition for such an algebra to be a Jordan algebra and give all corresponding isomorphism classes.

We study the shape identities arising in the theory of Bernstein algebras. We determine all shape identities of minimal degree for two subclasses of Bernstein algebras, namely, normal Bernstein algebras and exceptional Bernstein algebras.

It is known that the identities of lowest degree that are satisfied by all the Bernstein algebras are two identities of degree six. In this paper the authors study Bernstein algebras satisfying a polynomial identity of degree ≤4 that is not implied by the commutative law and they give a characterization of exceptional Bernstein algebras and normal...