Abstract
Let
be a ring with center
.
In this paper, we study the higher-order commutators with power central values on rings and algebras involving generalized derivations.
Motivated by [A. Alahmadi, S. Ali, A. N. Khan and M. Salahuddin Khan,
A characterization of generalized derivations on prime rings,
Comm. Algebra 44 2016, 8, 3201–3210],
we characterize generalized derivations and related maps that satisfy certain differential identities on prime rings.
Precisely, we prove that if a prime ring of characteristic different from two admitting generalized derivation
such that
for every
, then either
for every
or
satisfies
and
for every
and
, the Utumi quotient ring of
.
As an application, we prove that any spectrally generalized derivation on a semisimple Banach algebra satisfying the above mentioned differential identity must be a left multiplication map.