ABSTRACT
If an algebra satisfies the polynomial identity
(for short,
is
), then
is trivially Lie solvable of index
(for short,
is
). We prove that the converse holds for subalgebras of the upper triangular matrix algebra
any commutative ring, and
. We also prove that if a ring S is
(respectively,
), then the subring
of
comprising the upper triangular
matrices with constant main diagonal, is
(respectively,
) for all
. We also study two related questions, namely whether, for a field F, an
subalgebra of
for some n, with (F-)dimension larger than the maximum dimension
of a
subalgebra of
, exists, and whether a
subalgebra of
with (the mentioned) maximum dimension, other than the typical
subalgebras of
with maximum dimension, which were described by Domokos and refined by van Wyk and Ziembowski, exists. Partial results with regard to these two questions are obtained.
Notes
No potential conflict of interest was reported by the authors.