Communications in Algebra
®
, 35: 2647–2653, 2007
Copyright © Taylor & Francis Group, LLC
ISSN: 0092-7872 print/1532-4125 online
DOI: 10.1080/00927870701351278
EXAMPLES OF COMMUTATIVE RIGHT-NILALGEBRAS
OVER SMALL FIELDS
Antonio Behn
Department of Mathematics, Faculty of Science, University of Chile,
Santiago, Chile
Correa et al. (2003) proved that any commutative right-nilalgebra of nilindex 4
and dimension 4 is nilpotent in characteristic =2 3. They did not assume power-
associativity. In this article we will further investigate these algebras without the
assumption on the dimension and providing examples in those cases that are not covered
in the classification concentrating mostly on algebras generated by one element.
Key Words: Albert’s problem; Nilalgebras; Small characteristic.
1991 Mathematics Subject Classification: 17A30.
1. INTRODUCTION
In a nonassociative algebra A we define right principal powers of an element
x ∈ A recursively as follows:
x
1
= x x
n+1
= x
n
x
We say that A is power associative if for any x ∈ A, the subalgebra generated
by x is associative. A is right-nil if there is some n ∈ such that x
n
= 0 for all x ∈ A.
The smallest such n is called the right-nilindex of A.
We also define the following descending chains of subalgebras in A:
1. A
1
⊇ A
2
⊇ where A
1
= A and A
n
=
∑
r +s=n
A
r
A
s
for n> 1;
2. A
1
⊇ A
2
⊇ where A
1
= A and A
n
= A
n-1
A for n> 1;
3. A
1
⊇ A
2
⊇ where A
1
= A and A
n
= A
n-1
2
for n> 1.
When one of these chains goes to zero, we say that A is nilpotent, right-
nilpotent, or solvable, respectively. It is clear from the definition that nilpotent
implies right-nilpotent and right-nilpotent implies solvable.
If the right-nilindex of A is 2, there are two possibilities depending on the
characteristic of the underlying field. When char K = 2 we can easily see that the
multiplication is trivial. On the other hand, when char K = 2 we construct a power-
associative algebra A which is not solvable (Example 3.1).
Received January 9, 2006; Revised August 16, 2006. Communicated by A. Elduque.
Address correspondence to Antonio Behn, Department of Mathematics, Faculty of Science,
University of Chile, Casilla 653, Santiago, Chile; Fax: +56-2-2713882; E-mail: afbehn@gmail.com
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