# On centrally extended Jordan derivations and related maps in rings

B. Bhushan, G. S. Sandhu, S. Ali and D. Kumar  Department of Mathematics,
Faculty of Physical Sciences, Punjabi University, Patiala-147002, India.
<EMAIL_ADDRESS>Department of Mathematics, Patel Memorial National
College, Rajpura-140401, India<EMAIL_ADDRESS>Department of
Mathematics, Aligarh Muslim University, Aligarh-202002, India.
<EMAIL_ADDRESS>Department of Mathematics, Faculty of Physical
Sciences, Punjabi University, Patiala-147002, India<EMAIL_ADDRESS>
###### Abstract.

Let $R$ be a ring and $Z(R)$ be the center of $R.$ The aim of this paper is to
define the notions of centrally extended Jordan derivations and centrally
extended Jordan $\ast$-derivations, and to prove some results involving these
mappings. Precisely, we prove that if a $2$-torsion free noncommutative prime
ring $R$ admits a centrally extended Jordan derivation (resp. centrally
extended Jordan $\ast$-derivation) $\delta:R\to R$ such that

$[\delta(x),x]\in Z(R)~{}~{}(resp.~{}~{}[\delta(x),x^{\ast}]\in
Z(R))\text{~{}for~{}all~{}}x\in R,$

where ${}^{\prime}\ast^{\prime}$ is an involution on $R,$ then $R$ is an order
in a central simple algebra of dimension at most 4 over its center.

###### Key words and phrases:

Prime ring, semiprime ring, involution, centrally extended Jordan derivation,
centrally extended Jordan $\ast-$derivation
2020 Mathematical subject classification: 16W10, 16N60, 16W25

##  1\. Introduction and notions

Throughout this paper, $R$ denotes an associative ring with center $Z(R)$. The
maximal right ring of quotients of $R$ is denoted by $Q_{mr}(R)$ and the
center of $Q_{mr}(R)$ is called the extended centroid of $R$ and denoted by
$C,$ more information about these object can be found in [5]. For any $x,y\in
R$, the symbol $[x,y]$ (resp. $x\circ y$) denotes the commutator (resp. anti-
commutator) $xy-yx$ (resp. $xy+yx$). A ring $R$ is called _prime_ , if for any
$a,b\in R$, $aRb=(0)$ implies either $a=0$ or $b=0,$ and if $aRa=(0)$ implies
$a=0,$ then $R$ is called a _semiprime_ ring. For any $n\in\mathbb{Z}^{+},$
$R$ is called $n$-torsion free if $nx=0$ for all $x\in R,$ implies $x=0$. An
anti-automorphism ${}^{\prime}\ast^{\prime}$ of a ring $R$ is called
_involution_ if it is of period $2$. By a _ring with involution_ , we mean a
ring equipped with an involution ${}^{\prime}\ast^{\prime},$ it is also called
$\ast$-ring. Let $H(R):=\\{x\in R:x^{\ast}=x\\}$ and $S(R):=\\{x\in
R:x^{\ast}=-x\\};$ the elements of $H(R)$ are called _symmetric_ and the
elements of $S(R)$ are called _skew-symmetric_. Following Herstein [17, Ch.
6], $\overline{H(R)}$ will denote the ring generated by the symmetric elements
of $R$.

An additive mapping $d:R\to R$ is called a _derivation_ if $d(xy)=d(x)y+xd(y)$
for all $x,y\in R.$ For a fixed element $a\in R,$ a mapping $x\mapsto[a,x]$ is
called _inner derivation_ induced by ${}^{\prime}a^{\prime}$. An additive map
$d$ is called a _Jordan derivation_ if $d(x^{2})=d(x)x+xd(x)$ for all $x\in
R$. Obviously, every derivation is a Jordan derivation but the converse need
not be true (see [4, Example 3.2.1]). Moreover, the question that “when a
Jordan derivation is a derivation?” caused a new and significant area of
research (see [3, 9, 10, 18, 27]). In 1957, Herstein [18] showed that for
2-torsion free prime rings , every Jordan derivation is an ordinary
derivation. Later, Bre$\breve{s}$ar and Vukman [9] gave a brief and elegant
proof of this result. In the same year, Bre$\breve{s}$ar [10] showed that for
a rather wide class of rings, namely semiprime rings with 2-torsion free
condition, every Jordan derivation is a derivation. Thenceforth a considerable
amount of results has been proved in this direction.

Let $R$ be a $\ast$-ring. An additive mapping $d:R\to R$ is called a _$\ast$
-derivation_ if $d(xy)=d(x)y^{\ast}+xd(y)$ for all $x,y\in R$ and is called a
_Jordan $\ast$-derivation_ if $d(x^{2})=d(x)x^{\ast}+xd(x)$ for all $x\in R$.
The notions of $\ast$-derivation and Jordan $\ast$-derivation are first
mentioned in [11]. Note that the mapping $x\to ax^{\ast}-xa$, where $a$ is a
fixed element of $R$, is a Jordan $\ast$-derivation which is known as _inner
Jordan $\ast$-derivation_. The study of Jordan $\ast$-derivations has been
originated from the problem of representability of quadratic forms by bilinear
forms (see [25, 26]). Since then there has been a significant interest in the
study of algebraic structure of Jordan $\ast$-derivations in rings and
algebras, for a good cross-section we refer the reader to [2, 12, 21, 22]. For
further generalizations and recent results, see [13].

Let $S$ be a subset of $R,$ a mapping $\varphi$ is called centralizing (resp.
commuting) on $S$, if $[f(x),x]\in Z(R)$ (resp. $[f(x),x]=0$) for all $x\in
S$. The study of commuting and centralizing mappings goes back to 1955, when
Divinsky [16] proved that a simple artinian ring is commutative if it admits a
commuting nontrivial automorphism. In this line of investigation, Posner [24]
proved another remarkable result which states that if there exists a nonzero
centralizing derivation on $R,$ then $R$ must be commutative. Motivated by the
centralizing and commuting mappings, Ali and Dar [1] introduced
$\ast$-centralizing and $\ast$-commuting mappings and defined as follows: a
mapping $\varphi$ is called $\ast$-centralizing (resp. $\ast$-commuting) on a
set $S$ if $[f(x),x^{\ast}]\in Z(R)$ (resp. $[f(x),x^{\ast}]=0$) for all $x\in
S.$

There has been a rising literature on the investigation of centrally extended
mappings in rings under various settings; for e.g. see [6], [14], [15], [23].
Continuing in this line of investigation, in this paper we introduce centrally
extended Jordan derivations and give examples to show the existence of these
maps in a $2$-torsion free prime rings. We also show that there exists no
nonzero centrally extended Jordan derivation $\delta$ on a $2$-torsion free
noncommutative prime ring (resp. prime ring with involution
${}^{\prime}\ast^{\prime}$) satisfying $[\delta(x),x]\in Z(R)$ (resp.
$[\delta(x),x^{\ast}]\in Z(R)$) for all $x\in R,$ unless $R$ is an order in a
central simple algebra of dimension $4$ over its center. Finally, we give the
notion of centrally extended Jordan $\ast$-derivation and provide the
analogous studies.

##  2\. Preliminaries

By $s_{4},$ we denote the standard identity in four noncommuting variables,
which is defined as follows:

$s_{4}(x_{1},x_{2},x_{3},x_{4})=\sum_{\sigma\in
S_{4}}(-1)^{\sigma}x_{\sigma(1)}x_{\sigma(2)}x_{\sigma(3)}x_{\sigma(4)},$

where $S_{4}$ is the symmetric group of degree $4$ and $(-1)^{\sigma}$ is the
sign of permutation $\sigma\in S_{4}.$ It is known by the standard PI-theory
that, a prime ring $R$ satisfying $s_{4}$ can be characterized in a number of
ways, as follows:

######  Lemma 2.1.

[8, Lemma 1] Let $R$ be a prime ring with extended centroid $C$. Then the
following statements are equivalent:

  * (i)

$R$ satisfies $s_{4}.$

  * (ii)

$R$ is commutative or $R$ embeds into $M_{2}(F),$ for a field $F.$

  * (iii)

$R$ is algebraic of bounded degree $2$ over $C$ (i.e., for any $a\in R,$ there
exists a polynomial $x^{2}+\alpha x+\beta\in C[x]$ satisfied by $a$).

  * (iv)

$R$ satisfies $[[x^{2},y],[x,y]].$

######  Lemma 2.2.

[1, Lemma 2.2] Let $R$ be a $2$-torsion free semiprime ring with involution
${}^{\prime}\ast^{\prime}$. If an additive self-mapping $f$ of $R$ satisfies
$[f(x),x^{\ast}]\in Z(R)$ for all $x\in R,$ then $[f(x),x^{\ast}]=0$ for all
$x\in R.$

######  Lemma 2.3.

[5, Proposition 2.1.7 (ii)] Let $R$ be a prime ring, $Q_{mr}(R)$ be the
maximal right ring of quotients of $R$ and $\mathcal{D}$ be the set of all
right dense ideals of $R.$ Then for all $q\in Q_{mr}(R),$ there exists
$J\in\mathcal{D}$ such that $qJ\subseteq R.$

######  Lemma 2.4.

[7, Proposition 3.1] Let $R$ be a $2$-torsion free semiprime ring and $U$ be a
Jordan subring of $R.$ If an additive self-mapping $f:R\to R$ satisfies
$[f(x),x]\in Z(R)$ for all $x\in U,$ then $[f(x),x]=0$ for all $x\in U.$

######  Lemma 2.5.

[7, Theorem 3.2] Let $R$ be a prime ring. If an additive mapping $F:R\to R$ is
commuting on $R,$ then there exists $\lambda\in C$ and an additive $\xi:R\to
C,$ such that $F(x)=\lambda x+\xi(x)$ for all $x\in R.$

######  Lemma 2.6.

[17, Theorem 6.5.1] If $R$ is a semiprime ring, then $\overline{H(R)}$ is
semiprime, where $\overline{H(R)}$ is the ring generated by all symmetric
elements in $R$.

######  Lemma 2.7.

[17, Theorem 6.5.3] If $R$ is a semiprime ring, then
$Z(\overline{H(R)})\subseteq Z(R).$

######  Lemma 2.8.

[19, Lemma 2] If $R$ is a semiprime ring and $[S(R)^{2},S(R)^{2}]=(0),$ then
$R$ satisfies $s_{4}.$

######  Lemma 2.9.

[20, Theorem 3] Let $R$ be a prime ring with involution
${}^{\prime}\ast^{\prime}$ and center $Z(R).$ If $n$ be a fixed natural number
such that $x^{n}\in Z(R)$ for all $x\in H(R),$ then $R$ satisfies $s_{4}.$

######  Lemma 2.10.

[20, Theorem 7] Let $R$ be a prime ring with involution
${}^{\prime}\ast^{\prime}$ and center $Z(R).$ If $d$ is a nonzero derivation
on $R$ such that $d(x)x+xd(x)\in Z(R)$ for all $x\in S(R),$ then $R$ satisfies
$s_{4}.$

######  Lemma 2.11.

[27, Lemma 1.3] Let $R$ be a semiprime ring and $a\in R$ some fixed element.
If $a[x,y]=0$ for all $x,y\in R,$ then there exists an ideal $I$ of $R$ such
that $a\in I\subseteq Z(R)$ holds.

##  3\. Results on centrally extended Jordan derivations

This section deals with the study of centrally extended Jordan derivations of
rings. In fact, we characterize $2$-torsion free noncommutative prime rings
admitting $CE-$Jordan derivations. Recently, Bell and Daif [6] introduced
_centrally extended derivations_ which are obviously a generalization of
derivations; and they discussed the existence of these mappings in rings.
Accordingly, a self-mapping $d$ of $R$ is called a centrally extended
derivation if $d(x+y)-d(x)-d(y)\in Z(R)$ and $d(xy)-d(x)y-xd(y)\in Z(R)$ for
all $x,y\in R.$ Motivated by this, we now introduce a finer notion than
centrally extended derivation, and call a _centrally extended Jordan
derivation_ , as follows:

######  Definition 3.1.

A mapping $\delta:R\to R$ that satisfies

$\delta(x+y)-\delta(x)-\delta(y)\in Z(R),$ (A) $\delta(x\circ
y)-\delta(x)\circ y-x\circ\delta(y)\in Z(R)$ (B)

for all $x,y\in R,$ is called a _centrally extended Jordan derivation_ of $R$.
We shall abbreviate this map as $CE-$Jordan derivation.

######  Example 3.2.

Let $R=M_{2}(\mathbb{Z})\times\mathbb{Z}$ be a ring and define a mapping
$\delta:R\to R$ by

$\delta\left(\left(\begin{array}[]{cc}a&b\\\
c&d\end{array}\right),x\right)=\left(\left(\begin{array}[]{cc}0&-b\\\
c&0\end{array}\right),1\right).$

Then, it is straightforward to check that $\delta$ is a $CE-$Jordan derivation
of $R.$

######  Remark 3.3.

If $R$ is a 2-torsion free noncommutative prime ring, then it is not difficult
to see that an additive map $\delta:R\to R$ is a $CE-$Jordan derivation if and
only if $\delta(x^{2})-\delta(x)x-x\delta(x)\in Z(R)$ for all $x\in R.$ It is
natural to ask whether a $CE-$Jordan derivation can be a $CE-$derivation or a
Jordan derivation? In the following counter-example, we show that in case $R$
is a noncommutative prime ring, the above statement is not always true:
Let $\mathbb{Z}$ be the ring of integers and

$R=\left\\{\left(\begin{array}[]{cc}a&b\\\
c&d\end{array}\right)|~{}a,b,c,d\in\mathbb{Z}\right\\}$

be a noncommutative prime ring. Then a mapping $\delta:R\to R$ such that

$\delta\left(\begin{array}[]{cc}a&b\\\
c&d\end{array}\right)=\left(\begin{array}[]{cc}0&b\\\ b&0\end{array}\right)$

is a $CE-$Jordan derivation, but neither a $CE-$derivation nor a Jordan
derivation.

We begin with the following lemma:

######  Lemma 3.4.

Let $R$ be a 2-torsion free ring with no nonzero central ideal. If $\delta$ is
a $CE-$Jordan derivation of $R,$ then $\delta$ is additive.

###### Proof.

Let $\delta$ be a $CE-$Jordan derivation of $R$. In view of condition $(A),$
for any $x,y,z\in R,$ we have

$\delta(x+y)=\delta(x)+\delta(y)+c_{\delta(x,y,+)},$ (3.1)

where $c_{\delta(x,y,+)}\in Z(R).$ There exists some
$c_{\delta(z,x+y,\circ)}\in Z(R)$ such that

$\displaystyle\delta(z\circ(x+y))$ $\displaystyle=$
$\displaystyle\delta(z)\circ(x+y)+z\circ\delta(x+y)+c_{\delta(z,x+y,\circ)}$
(3.2) $\displaystyle=$ $\displaystyle\delta(z)\circ x+\delta(z)\circ
y+z\circ(\delta(x)+\delta(y)+c_{\delta(x,y,+)})+c_{\delta(z,x+y,\circ)}$
$\displaystyle=$ $\displaystyle\delta(z)\circ x+\delta(z)\circ
y+z\circ\delta(x)+z\circ\delta(y)+2zc_{\delta(x,y,+)}$
$\displaystyle+c_{\delta(z,x+y,\circ)}.$

Another way of looking at it is,

$\displaystyle\delta(z\circ(x+y))$ $\displaystyle=$
$\displaystyle\delta(z\circ x+z\circ y)$ (3.3) $\displaystyle=$
$\displaystyle\delta(z\circ x)+\delta(z\circ y)+c_{\delta(z\circ x,z\circ
y,+)}$ $\displaystyle=$ $\displaystyle\delta(z)\circ
x+z\circ\delta(x)+c_{\delta(z,x,\circ)}+\delta(z)\circ y+z\circ\delta(y)$
$\displaystyle+c_{\delta(z,y,\circ)}+c_{\delta(z\circ x,z\circ y,+)},$

where $c_{\delta(z\circ x,z\circ y,+)},c_{\delta(z,x,\circ)}$ and
$c_{\delta(z,y,\circ)}$ are the central elements.

Comparing (3.2) and (3.3), we conclude that
$2zc_{\delta(x,y,+)}+c_{\delta(z,x+y,\circ)}=c_{\delta(z\circ x,z\circ
y,+)}+c_{\delta(z,x,\circ)}+c_{\delta(z,y,\circ)}\in Z(R)$. It forces that
$Rc_{\delta(x,y,+)}\subseteq Z(R),$ where $c_{\delta(x,y,+)}$ is a fixed
central element in $R,$ but $R$ has no nonzero central ideal, therefore
$Rc_{\delta(x,y,+)}=(0).$ Likewise, we get $c_{\delta(x,y,+)}R=(0).$ It
implies that $c_{\delta(x,y,+)}\in A(R),$ the annihilator of $R.$ But $A(R)$
is always a central ideal in $R,$ hence our hypothesis forces $A(R)=(0)$ and
consequently $c_{\delta(x,y,+)}=0.$ From (3.1), we get
$\delta(x+y)=\delta(x)+\delta(y)$ for all $x,y\in R,$ as desired. ∎

######  Corollary 3.5.

Let $R$ be a 2-torsion free noncommutative prime ring. If $\delta$ is a
$CE-$Jordan derivation of $R,$ then $\delta$ is additive.

Now, we are in position to state and prove the first result of this paper.

######  Theorem 3.6.

Let $R$ be a $2$-torsion free noncommutative prime ring. If $R$ admits a
nonzero $CE-$Jordan derivation $\delta:R\to R$ such that $[\delta(x),x]\in
Z(R)$ for all $x\in R,$ then either $\delta=0$ or $R$ is an order in a central
simple algebra of dimension at most $4$ over its center.

###### Proof.

By the hypothesis, we have $[\delta(x),x]\in Z(R)$ for all $x\in R.$ In view
of Corollary 3.5, $\delta$ is additive and hence from Lemma 2.4, it follows
that

$[\delta(x),x]=0\mbox{~{}for~{}all~{}}x\in R.$ (3.4)

Since $\delta$ is an additive and commuting function, by Lemma 2.5, there
exists $\lambda\in C$ (extended centroid of $R$) and an additive mapping
$\sigma:R\to C$ such that

$\delta(x)=\lambda x+\sigma(x)\mbox{~{}for~{}all~{}}x\in R.$ (3.5)

Polarizing (3.4), we have

$[\delta(x),y]+[\delta(y),x]=0\mbox{~{}for~{}all~{}}x,y\in R.$

Replacing $y$ by $x\circ y,$ we get

$[\delta(x),x\circ y]+[\delta(x\circ y),x]=0\mbox{~{}for~{}all~{}}x,y\in R.$

It implies

$[\delta(x),x\circ y]+[\delta(x)\circ
y,x]+[x\circ\delta(y),x]=0\mbox{~{}for~{}all~{}}x,y\in R.$

Using (3.5) in the preceding relation to get

$[\lambda x+\sigma(x),x\circ y]+[(\lambda x+\sigma(x))\circ
y,x]+[x\circ(\lambda y+\sigma(y)),x]=0,$

that is

$[\lambda x,x\circ y]+[\lambda(x\circ y),x]+[\sigma(x)\circ
y,x]+[\lambda(x\circ y),x]+[x\circ\sigma(y),x]=0\mbox{~{}for~{}all~{}}x,y\in
R.$ (3.6)

It follows that $R$ satisfies

$\lambda[x,x\circ y]+\lambda[x\circ y,x]+2\sigma(x)[y,x]+\lambda[x\circ
y,x]=0,$ (3.7)

and so

$2\sigma(x)[y,x]+\lambda[x\circ y,x]=0\mbox{~{}for~{}all~{}}x,y\in R.$ (3.8)

Further, it implies

$2\sigma(x)[y,x]+\lambda[y,x^{2}]=0\mbox{~{}for~{}all~{}}x,y\in R,$

and a fortiori

$\lambda[[y,x^{2}],[y,x]]=0\mbox{~{}for~{}all~{}}x,y\in R.$

It implies that either $\lambda=0$ or $[[y,x^{2}],[y,x]]=0$ for all $x,y\in
R.$ By Lemma 2.1, the latter case is equivalent to the $s_{4}$ identity and
$R$ is assumed to be noncommutative, therefore $R$ is an order in a central
simple algebra of dimension at most 4 over $Z(R).$

On the other hand, let us assume that $\lambda=0.$ Then from (3.8), we have

$2\sigma(x)[y,x]=0\mbox{~{}for~{}all~{}}x,y\in R.$

Using the restriction on torsion of $R,$ we have

$\sigma(x)[y,x]=0\mbox{~{}for~{}all~{}}x,y\in R.$

Since $R$ is a prime ring, for each $x\in R,$ either $\sigma(x)=0$ or
$[R,x]=(0).$ Put $\mathfrak{U}=\\{x\in R:~{}\sigma(x)=0\\}$ and
$\mathfrak{V}=\\{x\in R:~{}[R,x]=(0)\\}.$ Therefore, we note that $R$ can be
written as the set-theoretic union of the additive subgroups $\mathfrak{U}$
and $\mathfrak{V},$ which is not possible. Thus, we have either
$R=\mathfrak{U}$ or $R=\mathfrak{V}.$ It implies that either $\sigma(x)=0$ for
all $x\in R$ or $[R,x]=(0)$ for all $x\in R$. If $\sigma(x)=0$ for all $x\in
R,$ then from (3.5), we find $\delta(x)=0$ for all $x\in R$. In the other case
$R$ is a commutative ring; which leads a contradiction. This completes the
proof. ∎

######  Theorem 3.7.

Let $R$ be a 2-torsion free noncommutative prime ring with involution
${}^{\prime}\ast^{\prime}$ that admits a $CE-$Jordan derivation $\delta:R\to
R$ such that $[\delta(x),x^{\ast}]\in Z(R)$ for all $x\in R.$ Then either
$\delta=0$ or $R$ is an order in a central simple algebra of dimension at most
$4$ over its center.

###### Proof.

Let us assume that $[\delta(x),x^{\ast}]\in Z(R)$ for all $x\in R.$ With the
aid of Corollary 3.5 and Lemma 2.2, we have

$[\delta(x),x^{\ast}]=0\mbox{~{}for~{}all~{}}x\in R.$ (3.9)

Applying involution in (3.9), we get

$[\delta(x)^{\ast},x]=0\mbox{~{}for~{}all~{}}x\in R.$ (3.10)

In view of Lemma 2.5, there exists $\lambda\in C$ and an additive mapping
$\sigma:R\to C$ such that

$\delta(x)^{\ast}=\lambda x+\sigma(x)\mbox{~{}for~{}all~{}}x\in R.$

It implies

$\delta(x)=\lambda^{\ast}x^{\ast}+\sigma(x)^{\ast}\mbox{~{}for~{}all~{}}x\in
R.$ (3.11)

We now split the proof into two parts, as follows:
Case 1. Suppose that there exists a non-zero element $c\in C$ such that
$c^{\ast}\neq c.$ Let $c^{\ast}-c=z_{c}.$ Clearly $z_{c}^{\ast}=-z_{c}\neq 0.$
Also $z_{c}\in C$. By Lemma 2.3, there exists a nonzero ideal $J$ of $R$ such
that $z_{c}J\subseteq R$. Polarizing (3.9), we get

$[\delta(x),y^{\ast}]+[\delta(y),x^{\ast}]=0\mbox{~{}for~{}all~{}}x,y\in R.$
(3.12)

Replacing $y$ by $y\circ r$ in (3.12), where $r\in J,$ we get

$[\delta(x),(y\circ r)^{\ast}]+[\delta(y\circ r),x^{\ast}]=0.$ (3.13)

Therefore, we have

$[\delta(x),y^{\ast}\circ r^{\ast}]+[\delta(y)\circ
r,x^{\ast}]+[y\circ\delta(r),x^{\ast}]=0\mbox{~{}for~{}all~{}}x,y\in R,r\in
J.$ (3.14)

In particular for $y=h\in H(R)$ in (3.14), we get

$[\delta(x),r^{\ast}\circ h]+[r\circ\delta(h),x^{\ast}]+[\delta(r)\circ
h,x^{\ast}]=0.$

Using (3.11) in the above expression, we obtain

$\lambda^{\ast}[x^{\ast},r^{\ast}\circ h]+\lambda^{\ast}[r^{\ast}\circ
h,x^{\ast}]+\lambda^{\ast}[r\circ
h,x^{\ast}]+2\sigma(r)^{\ast}[h,x^{\ast}]+2\sigma(h)^{\ast}[r,x^{\ast}]=0.$

It implies

$\lambda^{\ast}[r\circ
h,x^{\ast}]+2\sigma(r)^{\ast}[h,x^{\ast}]+2\sigma(h)^{\ast}[r,x^{\ast}]=0\mbox{~{}for~{}all~{}}x\in
R,r\in J,h\in H(R).$ (3.15)

Replacing $r$ by $z_{c}r=rz_{c}$ in (3.15), we get

$\lambda^{\ast}[r\circ
h,x^{\ast}]z_{c}+2\sigma(rz_{c})^{\ast}[h,x^{\ast}]+2\sigma(h)^{\ast}[r,x^{\ast}]z_{c}=0\mbox{~{}for~{}all~{}}x\in
R,r\in J,h\in H(R).$ (3.16)

Multiply (3.15) by $z_{c}$ and comparing with (3.16), we obtain

$(\sigma(rz_{c})^{\ast}-\sigma(r)^{\ast}z_{c})[h,x^{\ast}]=0.$

Therefore primeness of $R$ implies that either $[h,x^{\ast}]=0$ for all $h\in
H(R),x\in R$ or $\sigma(rz_{c})^{\ast}=\sigma(r)^{\ast}z_{c}$. In the former
case, $R$ satisfies $s_{4}$ identity by Lemma 2.9. But $R$ is noncommutative,
therefore $R$ is an order in a central simple algebra of dimension at most $4$
over its center. In the latter case, we have

$\sigma(rz_{c})^{\ast}=\sigma(r)^{\ast}z_{c}.$ (3.17)

Replacing $r$ by $rz_{c}$ in (3.14), we obtain

$[\delta(x),y^{\ast}\circ r^{\ast}]z_{c}^{\ast}+[\delta(y)\circ
r,x^{\ast}]z_{c}+[y\circ\delta(rz_{c}),x^{\ast}]=0.$

It implies

$-[\delta(x),y^{\ast}\circ
r^{\ast}]z_{c}+[y\circ\delta(rz_{c}),x^{\ast}]+[\delta(y)\circ
r,x^{\ast}]z_{c}=0\mbox{~{}for~{}all~{}}x,y\in R,r\in J.$ (3.18)

Multiplying (3.14) by $z_{c}$ and then adding it into (3.18), we find

$[y\circ\delta(rz_{c}),x^{\ast}]+[y\circ\delta(r),x^{\ast}]z_{c}+2[\delta(y)\circ
r,x^{\ast}]z_{c}=0\mbox{~{}for~{}all~{}}x,y\in R,r\in J,$

which by virtue of (3.11), leads to

$[y\circ(\lambda^{\ast}r^{\ast}z_{c}^{\ast}+\sigma(rz_{c})^{\ast}),x^{\ast}]+[y\circ(\lambda^{\ast}r^{\ast}+\sigma(r)^{\ast}),x^{\ast}]z_{c}+2[\delta(y)\circ
r,x^{\ast}]z_{c}=0.$

An application of (3.17) yields

$[y\circ(-\lambda^{\ast}r^{\ast}+\sigma(r)^{\ast}),x^{\ast}]z_{c}+[y\circ(\lambda^{\ast}r^{\ast}+\sigma(r)^{\ast}),x^{\ast}]z_{c}+2[\delta(y)\circ
r,x^{\ast}]z_{c}=0.$

It follows that

$2[y\circ\sigma(r)^{\ast},x^{\ast}]z_{c}+2[\delta(y)\circ
r,x^{\ast}]z_{c}=0\mbox{~{}for~{}all~{}}x,y\in R,r\in J.$

Since $z_{c}\neq 0,$ it implies that

$2(2\sigma(r)^{\ast}[y,x^{\ast}]+[r\circ\delta(y),x^{\ast}])=0\mbox{~{}for~{}all~{}}x,y\in
R,r\in J.$

Using 2-torsion freeness of $R$ and (3.11), we have

$2\sigma(r)^{\ast}[y,x^{\ast}]+[r\circ(\lambda^{\ast}y^{\ast}+\sigma(y)^{\ast}),x^{\ast}]=0.$

It implies

$2\sigma(r)^{\ast}[y,x^{\ast}]+\lambda^{\ast}[r\circ
y^{\ast},x^{\ast}]+2\sigma(y)^{\ast}[r,x^{\ast}]=0\mbox{~{}for~{}all~{}}x,y\in
R,r\in J.$ (3.19)

In particular, for $y\in J,$ we replace $y$ by $yz_{c}$ and get

$2\sigma(r)^{\ast}[y,x^{\ast}]z_{c}-\lambda^{\ast}[r\circ
y^{\ast},x^{\ast}]z_{c}+2\sigma(yz_{c})^{\ast}[r,x^{\ast}]=0.$

Using (3.17), we have

$2\sigma(r)^{\ast}[y,x^{\ast}]z_{c}-\lambda^{\ast}[r\circ
y^{\ast},x^{\ast}]z_{c}+2\sigma(y)^{\ast}[r,x^{\ast}]z_{c}=0.$ (3.20)

Multiplying (3.19) by $z_{c}$ and then subtract it from (3.20), we conclude
that

$\lambda^{\ast}[r\circ y^{\ast},x^{\ast}]z_{c}=0\mbox{~{}for~{}all~{}}x\in
R,~{}r,y\in J.$

By the primeness of $R$ we have either $[r\circ y,x]=0$ or $\lambda=0$. Since
$R$ is noncommutative, the first situation cannot occur; consequently
$\lambda=0.$ Thus $\delta(x)=\sigma(x)^{\ast}$ for all $x\in R.$ From relation
(B), we have

$\sigma(x\circ y)^{\ast}-\sigma(x)^{\ast}\circ y-x\circ\sigma(y)^{\ast}\in
Z(R)\mbox{~{}for~{}all~{}}x,y\in R,$

that is

$2\sigma(x)^{\ast}[y,x]=0\mbox{~{}for~{}all~{}}x,y\in R.$

Using $2$-torsion freeness hypothesis, we find

$\sigma(x)^{\ast}[y,x]=0\mbox{~{}for~{}all~{}}x,y\in R.$

Further proceeding as Theorem 3.6, we are done in this case.
Case 2. Let $c^{\ast}=c$ for all $c\in C.$ Polarizing (3.9), we have

$[\delta(x),y^{\ast}]+[\delta(y),x^{\ast}]=0\mbox{~{}for~{}all~{}}x,y\in R.$
(3.21)

Replacing $y$ by $h\circ k,$ where $k\in S(R)$ and $h\in H(R),$ we have

$-[\delta(x),h\circ k]+[\delta(h)\circ
k,x^{\ast}]+[h\circ\delta(k),x^{\ast}]=0.$ (3.22)

In view of (3.11), it follows that

$-[\lambda x^{\ast},h\circ k]+[(\lambda h+\sigma(h))\circ
k,x^{\ast}]+[h\circ(-\lambda k+\sigma(k)),x^{\ast}]=0.$

It implies

$-\lambda[x^{\ast},h\circ
k]+2\sigma(h)[k,x^{\ast}]+2\sigma(k)[h,x^{\ast}]=0\mbox{~{}for~{}all~{}}x\in
R,h\in H(R),k\in S(R).$ (3.23)

Replacing $x$ by $k$ in (3.23), we find

$\lambda[k^{2},h]-2\sigma(k)[h,k]=0\mbox{~{}for~{}all~{}}h\in H(R),k\in S(R).$
(3.24)

Taking involution on both side in (3.24) and using $z^{\ast}=z$ for all $z\in
C,$ we find

$\lambda[k^{2},h]+2\sigma(k)[h,k]=0\mbox{~{}for~{}all~{}}h\in H(R),k\in S(R).$
(3.25)

Adding (3.25) and (3.24), we obtain

$2\lambda[k^{2},h]=0\mbox{~{}for~{}all~{}}h\in H(R),k\in S(R).$ (3.26)

Thereby 2-torsion freeness and primeness of $R$ implies that either
$\lambda=0$ or $[k^{2},h]={0}$ for all $k\in S(R)$ and $h\in H(R).$ Latter
case implies $d_{h}(k)k+kd_{h}(k)=0$ for all $k\in S(R),$ where $d_{h}$ is the
inner derivation induced by $h.$ In view of Lemma 2.10, $R$ satisfies $s_{4}$
or $d_{h}=0.$ Therefore either $R$ satisfies $s_{4}$ or $H(R)\subseteq Z(R).$
By Lemma 2.9, in each of the situation, $R$ satisfies $s_{4}$. But $R$ is
noncommutative, hence $R$ is an order in a central simple algebra of dimension
at most $4$ over its center.

We now consider $\lambda=0.$ Using (B) and (3.11), we have

$\sigma(x\circ y)-\sigma(x)\circ y-x\circ\sigma(y)\in
Z(R)\mbox{~{}for~{}all~{}}x,y\in R.$

Therefore

$2\sigma(x)[y,x]=0\mbox{~{}for~{}all~{}}x,y\in R.$

Further proceeding as the proof of Theorem 3.6, we get the conclusion. ∎

##  4\. Results on centrally extended Jordan $\ast$-derivation

Let $R$ be a ring with involution ${}^{\prime}\ast^{\prime}.$ In [14], El-
Deken and Nabiel introduced the notion of _centrally extended
$\ast$-derivation_ and investigated the case when centrally extended
$\ast$-derivations are $\ast$-derivations. More specifically, they established
the following result: _If R is a semiprime $\ast$-ring with no nonzero central
ideals, then every centrally extended $\ast$-derivation $d$ on $R$ is a
$\ast$-derivation_. Motivated by the concept of centrally extended
$\ast$-derivations, we now introduce the notion of centrally extended Jordan
$\ast$-derivation as follows:

######  Definition 4.1.

A mapping $\delta:R\to R$ that satisfies

$\delta(x+y)-\delta(x)-\delta(y)\in Z(R),$ (C) $\delta(x\circ
y)-\delta(x)y^{\ast}-x\delta(y)-\delta(y)x^{\ast}-y\delta(x)\in Z(R)$ (D)

for all $x,y\in R$ is called _centrally extended Jordan $\ast$-derivation_ of
$R$. We shall abbreviate this map as $CE-$Jordan $\ast$-derivation.

######  Example 4.2.

Let $R=\bigg{\\{}\left(\begin{array}[]{ccc}0&a&b\\\ 0&0&c\\\ 0&0&0\\\
\end{array}\right):~{}a,b,c\in\mathbb{Z}\bigg{\\}}$ be a ring. Define mapping
$\delta,\ast:R\to R$ by

$\delta\left(\begin{array}[]{ccc}0&a&b\\\ 0&0&c\\\ 0&0&0\\\
\end{array}\right)=\left(\begin{array}[]{ccc}0&0&k\\\ 0&0&0\\\ 0&0&0\\\
\end{array}\right),\left(\begin{array}[]{ccc}0&a&b\\\ 0&0&c\\\ 0&0&0\\\
\end{array}\right)^{\ast}=\left(\begin{array}[]{ccc}0&c&b\\\ 0&0&a\\\ 0&0&0\\\
\end{array}\right)$

where $k$ is a fixed integer. Then, it is straight forward to check that
$\delta$ is a $CE-$Jordan $\ast$-derivation of $R,$ with involution
${}^{\prime}\ast^{\prime}$ of $R.$

######  Remark 4.3.

If $R$ is a 2-torsion free noncommutative prime ring with involution
${}^{\prime}\ast^{\prime}$, then an additive map $d$ is a $CE-$Jordan
$\ast$-derivation if and only if $d(x^{2})-d(x)x^{\ast}-xd(x)\in Z(R)$ for all
$x\in R.$ It is observed that, in this case, a $CE-$Jordan $\ast$-derivation
is not necessarily a Jordan $\ast$-derivation or $CE$ $\ast$-derivation, for
example:
Let $\mathbb{Z}$ be the ring of integers and
$R=\left\\{\left(\begin{array}[]{cc}a&b\\\
c&d\end{array}\right)|~{}a,b,c,d\in\mathbb{Z}\right\\},$ a prime ring. Define
a mapping $\ast:R\to R$ by $\left(\begin{array}[]{cc}a&b\\\
c&d\end{array}\right)^{\ast}=\left(\begin{array}[]{cc}d&-b\\\
-c&a\end{array}\right),$ which is an involution of $R$. If $\delta:R\to R$ be
a mapping such that $\delta\left(\begin{array}[]{cc}a&b\\\
c&d\end{array}\right)=\left(\begin{array}[]{cc}a&0\\\ 0&a\end{array}\right),$
then $\delta$ is a $CE-$Jordan $\ast$-derivation but neither Jordan
$\ast$-derivation nor $CE$ $\ast$-derivation.

In this section our focus will be on $CE-$Jordan $\ast$-derivations acting on
2-torsion free noncommutative prime rings. More specifically, we prove the
$CE-$Jordan $\ast$-derivation analogy of the above results. We begin our
discussions with the following lemma.

######  Lemma 4.4.

Let $R$ be a 2-torsion free semiprime ring with involution
${}^{\prime}\ast^{\prime}$ and with no nonzero central ideal. If $\delta$ is a
$CE-$Jordan $\ast$-derivation of $R,$ then $\delta$ is additive.

###### Proof.

For any $x,y,z\in R,$ in view of (C), it follows that there exists
$c_{\delta(x,y,+)}\in Z(R)$ such that

$\delta(x+y)=\delta(x)+\delta(y)+c_{\delta(x,y,+)},$ (4.1)

Using (D), we have

$\displaystyle\delta(z\circ(x+y))$ $\displaystyle=$
$\displaystyle\delta(z)(x^{\ast}+y^{\ast})+\delta(x+y)z^{\ast}+(x+y)\delta(z)$
(4.2) $\displaystyle+z\delta(x+y)+c_{\delta(z,x+y,\circ)}$ $\displaystyle=$
$\displaystyle\delta(z)(x^{\ast}+y^{\ast})+\bigg{(}\delta(x)+\delta(y)+c_{\delta(x,y,+)}\bigg{)}z^{\ast}+(x+y)$
$\displaystyle\delta(z)+z\bigg{(}\delta(x)+\delta(y)+c_{\delta(x,y,+)}\bigg{)}+c_{\delta(z,x+y,\circ)}$
$\displaystyle=$
$\displaystyle\delta(z)(x^{\ast}+y^{\ast})+(\delta(x)+\delta(y))z^{\ast}+(x+y)\delta(z)$
$\displaystyle+z(\delta(x)+\delta(y))+c_{\delta(x,y,+)}(z+z^{\ast})+c_{\delta(z,x+y,\circ)},$

where $c_{\delta(z,x+y,\circ)},c_{\delta(x,y,+)}$ are central elements.
Computing in a different manner, we get

$\displaystyle\delta(z\circ(x+y))$ $\displaystyle=$
$\displaystyle\delta(z\circ x+z\circ y)$ (4.3) $\displaystyle=$
$\displaystyle\delta(zx+xz)+\delta(zy+yz)+c_{\delta(z\circ x,z\circ y,+)}$
$\displaystyle=$
$\displaystyle\bigg{(}\delta(z)x^{\ast}+\delta(x)z^{\ast}+z\delta(x)+x\delta(z)+c_{\delta(z,x,\circ)}\bigg{)}+\bigg{(}\delta(z)y^{\ast}+\delta(y)z^{\ast}+z$
$\displaystyle\delta(y)+y\delta(z)+c_{\delta(z,y,\circ)}\bigg{)}+c_{\delta(z\circ
x,z\circ y,+)},$

where $c_{\delta(z\circ x,z\circ
y,+)},c_{\delta(z,x,\circ)},c_{\delta(z,y,\circ)}$ are central elements.

Comparing (4.2) and (4.3) to conclude
$(z+z^{\ast})c_{\delta(x,y,+)}+c_{\delta(z,x+y,\circ)}=c_{\delta(z,x,\circ)}+c_{\delta(z,y,\circ)}+c_{\delta(z\circ
x,z\circ y,+)}.$ It forces $(z+z^{\ast})c_{\delta(x,y,+)}$ for all $z\in R,$
where $c_{\delta(x,y,+)}$ is a fixed central element in $Z(R).$ Replacing $z$
by $h,$ where $h\in H(R)$, we have $2hc_{\delta(x,y,+)}\in Z(R).$ Using
2-torsion freeness hypothesis, we get

${}hc_{\delta(x,y,+)}\in Z(R)\mbox{~{}for~{}all~{}}h\in H(R).$ (4.4)

It implies

$c_{\delta(x,y,+)}[h,u]=0\mbox{~{}for~{}all~{}}h\in H(R),~{}u\in R.$

Replacing $h$ by $hh^{\prime},$ where $h^{\prime}\in H(R),$ we get

$c_{\delta(x,y,+)}[hh^{{}^{\prime}},u]=c_{\delta(x,y,+)}[h,u]h^{{}^{\prime}}+c_{\delta(x,y,+)}h[h^{{}^{\prime}},u]=0.$

Therefore, it follows that

$c_{\delta(x,y,+)}[\overline{H(R)},u]=(0)\mbox{~{}for~{}all~{}}u\in R,$ (4.5)

where $\overline{H(R)}$ is the ring generated by $H(R)$. In particular

$c_{\delta(x,y,+)}[\overline{H(R)},\overline{H(R)}]=(0).$

In view of Lemma 2.6, $\overline{H(R)}$ is also a semi prime ring. Now, we
shall first show that $c_{\delta(x,y,+)}\in H(R).$ Suppose that
$c_{\delta(x,y,+)}\not\in H(R),$ i.e., $c_{\delta(x,y,+)}^{\ast}\neq
c_{\delta(x,y,+)}.$ Then $(c_{\delta(x,y,+)}^{\ast}-c_{\delta(x,y,+)})k\in
H(R)$ for all $k\in S(R)$. Using (4.4), we get

$c_{\delta(x,y,+)}(c_{\delta(x,y,+)}^{\ast}-c_{\delta(x,y,+)})k\in
Z(R)\mbox{~{}for~{}all~{}}k\in S(R).$ (4.6)

Applying involution, we have

$c_{\delta(x,y,+)}^{\ast}(c_{\delta(x,y,+)}^{\ast}-c_{\delta(x,y,+)})k\in
Z(R)\mbox{~{}for~{}all~{}}k\in S(R).$ (4.7)

Subtracting $(\ref{eq-32})$ from $(\ref{eq-33})$, we get

$(c_{\delta(x,y,+)}^{\ast}-c_{\delta(x,y,+)})^{2}k\in
Z(R)\mbox{~{}for~{}all~{}}k\in S(R).$ (4.8)

Using (4.4), we find $c_{\delta(x,y,+)}^{\ast}h\in Z(R)$ for all $h\in H(R)$.
An easy consequence of this is

$(c_{\delta(x,y,+)}^{\ast}-c_{\delta(x,y,+)})h\in
Z(R)\mbox{~{}for~{}all~{}}h\in H(R).$ (4.9)

Since $c_{\delta(x,y,+)}\in Z(R)$, left multiplying (4.9) by
$c_{\delta(x,y,+)}^{\ast}-c_{\delta(x,y,+)}$ in order to find

$(c_{\delta(x,y,+)}^{\ast}-c_{\delta(x,y,+)})^{2}h\in
Z(R)\mbox{~{}for~{}all~{}}h\in H(R).$ (4.10)

Adding $(\ref{eq-34})$ and $(\ref{eq-36})$, we have

$(c_{\delta(x,y,+)}^{\ast}-c_{\delta(x,y,+)})^{2}(h+k)\in
Z(R)\mbox{~{}for~{}all~{}}h\in H(R),~{}k\in S(R).$ (4.11)

It yields

$(c_{\delta(x,y,+)}^{\ast}-c_{\delta(x,y,+)})^{2}R\subseteq Z(R).$

Since $R$ does not contain non-zero central ideals, thus we have
$(c_{\delta(x,y,+)}^{\ast}-c_{\delta(x,y,+)})^{2}=0,$ where
$c_{\delta(x,y,+)}^{\ast}-c_{\delta(x,y,+)}\in Z(R)$. The fact that center of
a semiprime ring has no nonzero nilpotent elements, forces
$c_{\delta(x,y,+)}^{\ast}-c_{\delta(x,y,+)}=0$, a contradiction to our
assumption. Hence $c_{\delta(x,y,+)}\in H(R).$

Therefore in view of Lemma 2.11, our expression
$c_{\delta(x,y,+)}[\overline{H(R)},\overline{H(R)}]=(0)$ implies that there
exists an ideal $I\subseteq\overline{H(R)}$ such that $c_{\delta(x,y,+)}\in
I\subseteq Z(\overline{H(R)})$. Using Lemma 2.7, we get $c_{\delta(x,y,+)}\in
I\subseteq Z(R)$. As $R$ does not contain non-zero central ideals, we get
$c_{\delta(x,y,+)}=0;$ and hence $\delta$ is additive, by (4.1). It completes
the proof. ∎

######  Corollary 4.5.

Let $R$ be a 2-torsion free noncommutative prime ring with involution
${}^{\prime}\ast^{\prime}$. If $\delta$ is a $CE-$Jordan $\ast$-derivation of
$R,$ then $\delta$ is additive.

######  Theorem 4.6.

Let $R$ be a 2-torsion free noncommutative prime ring with involution
${}^{\prime}\ast^{\prime}$ that admits a $CE-$Jordan $\ast$-derivation
$\delta:R\to R$ such that $[\delta(x),x]\in Z(R)$ for all $x\in R.$ Then
either $\delta=0$ or $R$ is an order in a central simple algebra of dimension
at most $4$ over its center.

###### Proof.

Let us assume that $[\delta(x),x]\in Z(R)$ for all $x\in R.$ By Lemma 2.4 and
Lemma 4.4, we conclude that

$[\delta(x),x]=0\mbox{~{}for~{}all~{}}x\in R.$ (4.12)

Polarizing (4.12), we have

$[\delta(x),y]+[\delta(y),x]=0\mbox{~{}for~{}all~{}}x,y\in R.$ (4.13)

Replacing $y$ by $k^{2},$ where $k\in S(R)$ in (4.13), we arrive at

$[\delta(x),k^{2}]+[-\delta(k)k+k\delta(k),x]=0\mbox{~{}for~{}all~{}}x\in
R,\;k\in S(R).$ (4.14)

Since $\delta$ is a commuting and additive function thereby using Lemma 2.5,
there exists $\lambda\in C$ and an additive mapping $\sigma:R\to C$ such that

$\delta(x)=\lambda x+\sigma(x)\mbox{~{}for~{}all~{}}x\in R.$ (4.15)

With this, it follows from (4.14) that

$[\lambda x,k^{2}]+[-(\lambda k+\sigma(k))k+k(\lambda
k+\sigma(k)),x]=0\mbox{~{}for~{}all~{}}x\in R,\;k\in S(R).$

In fact, we have

$\lambda[x,k^{2}]=0\mbox{~{}for~{}all~{}}x\in R,\;k\in S(R).$ (4.16)

Since the center of a prime ring contains no proper zero divisors, it follows
that either $\lambda=0$ or $k^{2}\subseteq Z(R)\mbox{~{}for~{}all~{}}k\in
S(R)$. In the latter case, we have $0=[k^{2},x]=[k,x]k+k[k,x]$ for all $x\in
R$ and $k\in S(R).$ It can be seen as $d_{x}(k)k+kd_{x}(k)=0$ for all $k\in
S(R)$, where $d_{x}(k)=[k,x]$ is the inner derivation induced by $x\in R$. In
view of Lemma 2.10, either $R$ satisfies $s_{4}$ or $d_{x}(t)=0$ for all $t\in
R$. Clearly, $d_{x}(t)=[x,t]=0$ for all $x,t\in R$ leads a contradiction. Thus
we conclude that $R$ is an order in a central simple algebra of dimension at
most $4$ over its center.

Now, if $\lambda=0,$ then from (4.15), we have $\delta(x)=\sigma(x)$ for all
$x\in R$. In view of $(D)$ and torsion condition on $R,$ it implies

$\sigma(x^{2})-\sigma(x)x^{\ast}-x\sigma(x)\in Z(R)\mbox{~{}for~{}all~{}}x\in
R,$

and so

$\sigma(x)[x+x^{\ast},y]=0\mbox{~{}for~{}all~{}}x,y\in R.$

Using Brauer’s trick, we have either $\sigma(x)=0$ for all $x\in R$ or
$x+x^{\ast}\in Z(R)$ for all $x\in R$. If $\sigma=0,$ then by (4.15)
$\delta=0,$ as desired. Now if $x+x^{\ast}\in Z(R)$ for all $x\in R,$ then
replacing $x$ by $h\in H(R),$ we get $H(R)\subseteq Z(R).$ Hence by Lemma 2.9,
we are done. ∎

######  Theorem 4.7.

Let $R$ be a $2$-torsion free noncommutative prime ring with involution
${}^{\prime}\ast^{\prime}$ that admits a $CE-$Jordan $\ast$-derivation
$\delta:R\to R$ such that $[\delta(x),x^{\ast}]\in Z(R)$ for all $x\in R.$
Then either $\delta=0$ or $R$ is an order in a central simple algebra of
dimension at most $4$ over its center.

###### Proof.

Let us consider $[\delta(x),x^{\ast}]\in Z(R)$ for all $x\in R.$ In light of
Lemma 2.2 and Lemma 4.4, we may infer that

$[\delta(x),x^{\ast}]=0\mbox{~{}for~{}all~{}}x\in R.$ (4.17)

This implies that

$[\delta(x)^{\ast},x]=0\mbox{~{}for~{}all~{}}x\in R.$ (4.18)

Using Lemma 2.5 in Eq. (4.18), there exists $\lambda\in C$ and an additive
mapping $\sigma:R\to C$ such that

$\delta(x)^{\ast}=\lambda x+\sigma(x)\mbox{~{}for~{}all~{}}x\in R,$

and equivalently we have

$\delta(x)=\lambda^{\ast}x^{\ast}+\sigma(x)^{\ast}\mbox{~{}for~{}all~{}}x\in
R.$ (4.19)

Polarizing (4.17), we find

$[\delta(x),y^{\ast}]+[\delta(y),x^{\ast}]=0\mbox{~{}for~{}all~{}}x,y\in R.$

Replacing $y$ by $k^{2}$ in the last expression, where $k\in S(R),$ we get

$[\delta(x),k^{2}]+[-\delta(k)k+k\delta(k),x^{\ast}]=0.$

Using (4.19), it yields

$[\lambda^{\ast}x^{\ast},k^{2}]+[-(\lambda^{\ast}k^{\ast}+\sigma(k))k+k(\lambda^{\ast}k^{\ast}+\sigma(k)),x^{\ast}]=0\mbox{~{}for~{}all~{}}x\in
R,~{}k\in S(R).$ (4.20)

Since $\lambda\in C$ and $\sigma(R)\subseteq C,$ we obtain

$\lambda^{\ast}[x^{\ast},k^{2}]=0\mbox{~{}for~{}all~{}}x\in R,~{}k\in S(R).$

Further, a similar demonstration that is given in the proof of Theorem 4.6
ensures the conclusion. ∎

## Conflict of interest

Here we declare that there is no conflict of interest in this work.

## References

  * [1] S. Ali, N. A. Dar, On $\ast$-centralizing mappings in rings with involution, Georgian Math. J., 21 (1) (2014), 25-28.
  * [2] S. Ali, A. Fo$\check{s}$ner, Maja Fo$\check{s}$ner, M. S. Khan, On generalized Jordan triple $(\alpha,\beta)^{\ast}$-derivations and related mappings, Mediterr. J. Math., 10 (2013), 1657-1668.
  * [3] M. Ashraf, N. Rehman, On Jordan generalized derivations in rings, Math. J. Okayama Univ., 42 (2000), 7–9.
  * [4] M. Ashraf, S. Ali, C. Haetinger, On derivations in rings and their applications, The Aligarh Bull. Math., 25 (2) (2006), 79-107.
  * [5] K. I. Beidar, W. S. Martindale III, A. V. Mikhalev, Rings with Generalized Identities, Pure Appl. Math. 196, Marcel Dekker Inc., New York, 1996.
  * [6] H. E. Bell, M. N. Daif, On centrally-extended maps on rings, Beitr. Algebra Geom., 57 (2016), 129-136.
  * [7] M. Bre$\breve{s}$ar, Centralizing mappings and derivations in prime rings, J. Algebra, 156 (2) (1993), 385-394.
  * [8] M. Bre$\breve{s}$ar, Commuting traces of biadditive mappings, commutativity-preserving mappings and Lie mappings, Trans. Amer. Math. Soc., 335 (2) (1993), 525-546.
  * [9] M. Bre$\breve{s}$ar, J. Vukman, Jordan derivations on prime rings, Bull. Austral. Math. Soc., 37 (1988), 321-322.
  * [10] M. Bre$\breve{s}$ar, Jordan derivations on semiprime rings, Proc. Amer. Math. Soc., 104 (4) (1988), 1003-1006.
  * [11] M. Bre$\breve{s}$ar, J. Vukman, On some additive mappings in rings with involution, Aequationes Math., 38 (1989), 178-185.
  * [12] M. Bre$\breve{s}$ar, B. Zalar, On the structure of Jordan $\ast$-derivations, Colloq. Math., 63 (1992), 163-171.
  * [13] N. A. Dar, S. Ali, On the structure of generalized Jordan $\ast$-derivations of prime rings, Commun. Algebra, 49 (4) (2021), 1422-1430.
  * [14] S. F. El-Deken, H. Nabiel, Centrally-extended generalized $\ast$-derivations on rings with involution, Beitr Algebra Geom., 60 (2019), 217-224.
  * [15] S. F. El-Deken, M. M. El-Soufi, On centrally extended reverse and generalized reverse derivations, Indian J. Pure Appl. Math., 51 (3) (2020), 1165-1180.
  * [16] N. Divinsky, On commuting automorphisms of rings, Trans. Royal Soc. Can. Sec. III, 3 (49) (1955), 19-22.
  * [17] I. N. Herstein, Rings with involution, University of Chicago Press, Chicago, 1976.
  * [18] I. N. Herstein, Jordan derivations of prime rings, Proc. Amer. Math. Soc., 8 (1957), 1104-1119.
  * [19] C. Lanski, Lie structure in semi-prime rings with involution, Commun. Algebra, 4 (8) (1976), 731-746.
  * [20] T. K. Lee, P. H. Lee, Derivations centralizing symmetric or skew elements, Bull. Inst. Math. Acad. Sinica, 14 (3) (1986), 249-256.
  * [21] T. K. Lee, Y. Zhou, Jordan $\ast$-derivations of prime rings, J. Algebra Appl., 13 (4) (2014) 1350126 (9 pages).
  * [22] T. K. Lee, T. L. Wong, Y. Zhou, The structure of Jordan $\ast$-derivations of prime rings, Linear Multilinear Algebra, 63 (2) (2015), 411-422.
  * [23] N. Muthana, Z. Alkhmisi, On centrally-extended multiplicative (generalized)-($\alpha,$ $\beta$)-derivations in semiprime rings, 49 (2) (2020), 578-585.
  * [24] E. C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc., 8 (1957), 1093-1100.
  * [25] P. $\check{S}$emrl, On Jordan $\ast$-derivations and an application, Colloq. Math., 59 (1990), 241-251.
  * [26] P. $\check{S}$emrl, Quadratic functionals and Jordan $\ast$-derivations, Stud. Math., 97 (1991), 157-165.
  * [27] B. Zalar, On centralisers of semiprime rings, Comment. Math. Univ. Carolinae, 32 (4) (1991), 609-614.