Varieties of right-commutative algebras defined by identity a(aa)=0

Article

Abstract

An algebra R is called a right-commutative algebra if it satisfies the right-commutative identity
(x1x2)x3 − (x1x3)x2 = 0,
for all elements x1, x2, x3 ∈ R.
This algebra is also well known by another name, for example, NAP algebra or a non-associative
permutative algebra. Varieties of right-commutative algebras contain important subvarieties such as Novikov
and bicommutative algebras.
In this paper, we consider two subvarieties M and N of the variety of right-commutative algebras. The
subvariety M is defined by the identities x1(x1x1) = 0, x1[x1, x2] = 0 and x1(x2x1) − x2(x1x1) = 0.
The subvariety N is defined by the identities x1(x1x1) = 0, x1(x1x2) − 2x1(x2x1) + x2(x1x1) = 0 and
[x1, x2]x1 = 0.
We study the free algebras in the varieties M and N of right-commutative algebras. We investigate the
Sn-module structures of the free algebras Fn(M) and Fn(N) in the varieties M and N, respectively. As a
consequence, we prove that the subvarieties M and N of the variety of right-commutative algebras satisfy
the distributivity conditions. Moreover, we describe all subvarieties of the varieties M and N.