FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2011/2012, VOLUME 17, NUMBER 1, PAGES 127-141

**Almost primitive elements of free nonassociative algebras of small
ranks**

A. V. Klimakov

A. A. Mikhalev

Abstract

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Let $K$ be
a field, $X\; =\; \{x$_{1},
¼
,x_{n}}, and let $F(X)$ be the free
nonassociative algebra over the field $K$ with the
set $X$ of
free generators.
A. G. Kurosh proved that subalgebras of free nonassociative
algebras are free.
A subset $M$ of nonzero elements of
the algebra $F(X)$ is said to be
primitive if there is a set $Y$ of free generators of
$F(X)$,
$F(X)\; =\; F(Y)$, such
that $M$Í Y (in this case we
have $|Y|\; =\; |X|\; =\; n$).
A nonzero element $u$ of the free algebra
$F(X)$ is said to
be an almost primitive if $u$ is not a primitive
element of the algebra $F(X)$, but $u$ is a primitive
element of any proper subalgebra of $F(X)$ that contains it.
In this article, for free nonassociative algebras of rank $1$ and $2$ criteria for homogeneous
elements to be almost primitive are obtained and algorithms to
recognize homogeneous almost primitive elements are constructed.
New examples of almost primitive elements of free nonassociative
algebras of rank $3$ are constructed.

Location: http://mech.math.msu.su/~fpm/eng/k1112/k111/k11107h.htm

Last modified: January 31, 2012