Let $\mathscr{A}$ be a ring with its center $\mathscr{Z}(\mathscr{A}).$ An additive mapping $\xi\colon \mathscr{A}\to \mathscr{A}$ is called a homoderivation on $\mathscr{A}$ if
$\forall\ a,b\in ...\mathscr{A}\colon\quad \xi(ab)=\xi(a)\xi(b)+\xi(a)b+a\xi(b).$
An additive map $\psi\colon \mathscr{A}\to \mathscr{A}$ is called a generalized homoderivation with associated homoderivation $\xi$ on $\mathscr{A}$ if
$\forall\ a,b\in \mathscr{A}\colon\quad\psi(ab)=\psi(a)\psi(b)+\psi(a)b+a\xi(b).$
This study examines whether a prime ring $\mathscr{A}$ with a generalized homoderivation $\psi$ that fulfils specific algebraic identities is commutative. Precisely, we discuss the following identities:
$\psi(a)\psi(b)+ab\in \mathscr{Z}(\mathscr{A}),\quad\psi(a)\psi(b)-ab\in \mathscr{Z}(\mathscr{A}),\quad\psi(a)\psi(b)+ab\in \mathscr{Z}(\mathscr{A}),$
$\psi(a)\psi(b)-ab\in \mathscr{Z}(\mathscr{A}),\quad\psi(ab)+ab\in \mathscr{Z}(\mathscr{A}),\quad\psi(ab)-ab\in \mathscr{Z}(\mathscr{A}),$
$\psi(ab)+ba\in \mathscr{Z}(\mathscr{A}),\quad\psi(ab)-ba\in \mathscr{Z}(\mathscr{A})\quad (\forall\ a, b\in \mathscr{A}).$
Furthermore, examples are given to prove that the restrictions imposed on the hypothesis of the various theorems were not superfluous.
Abstract The main purpose of this paper is to scrutinize the deportment of generalized derivations of R satisfying some functional * {*} -identities involving the center of the factor ring R / P ...{R/P} where P is a prime ideal of the ring R . Moreover, we suggest to give generalization of some well known results.
The major goal of this paper is to study the commutativity of prime rings with involution that meet specific identities using left centralizers. The results obtained in this paper are the ...generalization of many known theorems. Finally, we provide some examples to show that the conditions imposed in the hypothesis of our results are not superfluous.
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Let R be a prime ring of characteristic not 2 and U be a noncentral square closed Lie ideal of R. An additive mapping Hon R is called a homoderivation if H(xy) =H(x)H(y)+H(x)y+xH(y)for all x, y∈R. In ...this paper we investigate homoderivations satisfying certain differential identitieson square closed Lie ideals of prime rings.
Abstract
Let
R
be a prime ring with involution
⋆
{\star}
, and let σ, τ be endomorphisms on
R
. For any
x
,
y
∈
R
{x,y\in R}
, let
(
x
,
y
)
σ
,
τ
=
x
σ
(
y
)
+
τ
(
y
)
x
...{(x,y)_{\sigma,\tau}=x\sigma(y)+\tau(y)x}
and
C
σ
,
τ
(
R
)
=
{
x
∈
R
∣
x
σ
(
y
)
=
τ
(
y
)
x
}
{C_{\sigma,\tau}(R)=\{x\in R\mid x\sigma(y)=\tau(y)x\}}
.
An additive subgroup
U
of
R
is said to be a
(
σ
,
τ
)
{(\sigma,\tau)}
-right Jordan ideal (resp.
(
σ
,
τ
)
{(\sigma,\tau)}
-left
Jordan ideal) of
R
if
(
U
,
R
)
σ
,
τ
⊆
U
{(U,R)_{\sigma,\tau}\subseteq U}
(resp.
(
R
,
U
)
σ
,
τ
⊆
U
{(R,U)_{\sigma,\tau}\subseteq U}
), and
U
is called a
(
σ
,
τ
)
{(\sigma,\tau)}
-Jordan ideal if
U
is both a
(
σ
,
τ
)
{(\sigma,\tau)}
-right
Jordan ideal and a
(
σ
,
τ
)
{(\sigma,\tau)}
-left Jordan ideal of
R
. A
(
σ
,
τ
)
{(\sigma,\tau)}
-Jordan ideal
U
of
R
is said to be a
(
σ
,
τ
)
{(\sigma,\tau)}
-
⋆
{\star}
-Jordan ideal if
U
⋆
=
U
{U^{\star}=U}
.
In the present paper, it is shown that if
U
is commutative, then
R
is commutative. The commutativity of
R
is also obtained if
(
U
,
U
)
σ
,
τ
⊆
C
σ
,
τ
(
R
)
{(U,U)_{\sigma,\tau}\subseteq C_{\sigma,\tau}(R)}
. Some more results are obtained on the
⋆
{\star}
-prime ring with a characteristic different from 2.
On n-Jordan homoderivations in rings Belkadi, Said; Ali, Shakir; Taoufiq, Lahcen
Georgian mathematical journal,
02/2024, Volume:
31, Issue:
1
Journal Article
Peer reviewed
Let
be a ring and let
be a fixed integer. An additive mapping
of a ring
into itself is called an
-Jordan homoderivation if
holds for all
. In this paper, we initiate the study of
-Jordan ...homoderivations on rings. Precisely, we characterize
-Jordan homoderivations in terms of homoderivations and anti-homomorphisms under certain conditions. Finally, we conclude our paper with a direction for further research.
Let R be a noncommutative prime ring equipped with an antiautomorphism f such that (i) f(x
k
), x
m
= 0, all x
R, where k, m are fixed positive integers. In this article, we establish that from (i) ...follows (ii) f(x), x = 0, all x
R. More is to be said about the type of f and the structure of R.
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