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
  • General Mathematics

Abstract

Let

{\mathcal{R}}
be a prime ring of characteristic not equal to 2, let
𝒰 {\mathcal{U}}
be Utumi quotient ring of
{\mathcal{R}}
and let
𝒞 {\mathcal{C}}
be the extended centroid of
{\mathcal{R}}
. Let Δ be a generalized derivation on
{\mathcal{R}}
, and let
δ 1 {\delta_{1}}
and
δ 2 {\delta_{2}}
be derivations on
{\mathcal{R}}
. Let
p ( v ) {p(v)}
be a multilinear polynomial on
{\mathcal{R}}
, which is non-central valued on
{\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 n {v\in\mathcal{R}^{n}}
, then we find the complete description of Δ,
δ 1 {\delta_{1}}
and
δ 2 {\delta_{2}}
.