Abstract
Let R be a ring, Q its symmetric Martindale quotient ring, C its extended centroid, I a nonzero ideal of R and F a generalized derivation with associated non-zero derivation d of R,
and
fixed integers. Let
be a non-zero multilinear polynomial over C in t non-commuting variables,
be any subset of R and
. We prove the following results:
If R is prime and
for all
, then
is central valued on R.
If R is prime and
, for all
, then
is power central valued on R, unless
.
If R is semiprime and
for all
, then
, for any
and
, that is there exists a central idempotent element
such that
, d vanishes identically on eQ and
is central valued on
.
If R is semiprime and
is zero or invertible in R, for all
, then either R is a division ring or it is the ring of 2 × 2 matrices over a division ring, unless when
, for any
and
.
If R is prime and I is a non-zero right ideal of R such that
and
, for all
, then
is an identity on I.
Let R be prime and I a non-zero right ideal of R such that
and
, for all
. If there exists
such that
, then either
is power central valued on R or
is an identity on I, unless
.