DOI: 10.1515/gmj-2023-2109 ISSN: 1072-947X
Centralizing identities involving generalized derivations in prime rings
Vincenzo De Filippis, Pallavee Gupta, Shailesh Kumar Tiwari, Balchand Prajapati Abstract
Let
ℛ
{\mathcal{R}}
be a prime ring of characteristic not equal to 2, let
𝒰
{\mathcal{U}}
be Utumi quotient ring of
ℛ
{\mathcal{R}}
and let
𝒞
{\mathcal{C}}
be the extended centroid of
ℛ
{\mathcal{R}}
. Let Δ be a generalized derivation on
ℛ
{\mathcal{R}}
, and let
δ
1
{\delta_{1}}
and
δ
2
{\delta_{2}}
be derivations on
ℛ
{\mathcal{R}}
. Let
p
(
v
)
{p(v)}
be a multilinear polynomial on
ℛ
{\mathcal{R}}
, which is non-central valued on
ℛ
{\mathcal{R}}
. If
δ
1
(
Δ
2
(
p
(
v
)
)
p
(
v
)
)
=
δ
2
(
Δ
(
p
(
v
)
2
)
)
{\delta_{1}(\Delta^{2}(p(v))p(v))=\delta_{2}(\Delta(p(v)^{2}))}
for all
v
∈
ℛ
n
{v\in\mathcal{R}^{n}}
, then we find the complete description of Δ,
δ
1
{\delta_{1}}
and
δ
2
{\delta_{2}}
.