On Generalized $(m, n)$-Jordan Derivations and Centralizers of Semiprime Rings

TL;DR

This paper confirms two conjectures by proving that in semiprime rings, generalized $(m,n)$-Jordan derivations and centralizers are actually derivations and centralizers, under certain conditions, clarifying their structure.

Contribution

It provides an affirmative proof that generalized $(m,n)$-Jordan derivations and centralizers are standard derivations and centralizers in semiprime rings, resolving previous conjectures.

Findings

Generalized $(m,n)$-Jordan derivations are derivations under certain conditions.

Generalized $(m,n)$-Jordan centralizers are two-sided centralizers under certain conditions.

Results apply to semiprime rings with specific torsion restrictions.

Abstract

In this paper we give an affirmative answer to two conjectures on generalized $(m,n)$-Jordan derivations and generalized $(m,n)$-Jordan centralizers raised in [S. Ali and A. Fo\v{s}ner, \textit{On Generalized $(m, n)$-Derivations and Generalized $(m, n)$-Jordan Derivations in Rings,} Algebra Colloq. \textbf{21} (2014), 411--420] and [A. Fo\v{s}ner, \textit{A note on generalized $(m,n)$-Jordan centralizers,} Demonstratio Math. \textbf{46} (2013), 254--262]. Precisely, when $R$ is a semiprime ring, we prove, under some suitable torsion restrictions, that every nonzero generalized $(m,n)$-Jordan derivation (resp., a generalized $(m,n)$-Jordan centralizer) is a derivation (resp., a two-sided centralizer).

Equations

$$(m+n)d(xy)=2md(x)y+2nxd(y)$$

$$(m+n)D(xy)=2mD(x)y+2nxd(y)$$

$$(m+n)d(x^2)=2md(x)x+2nxd(x)$$

$$(m+n)F(x^2)=2mF(x)x+2nxd(x)$$

$$(m+n)T(x^2)=mT(x)x+nxT(x)$$

$$(m+n)T(x^2)=mT(x)x+nx{T}_0(x)$$

$$(m+n)D(x^2)=2mD(x)x,x\in R.$$

$$(m+n)D(x^4)=2mD(x^2)x^2,x\in R.$$

$$(m+n{)}^{2}D(x^4)=4m^{2}D(x)x^3,x\in R.$$

$$(m+n{)}^{2}D(x^4)=m(n-m)D(x)x^3+m(3m+n)D(x)x^3+m(m-n)D(x^2)x^2,x\in R.$$

$$2(m+n{)}^{2}D(x^4)=2m(n-m)D(x)x^3+2m(3m+n)D(x)x^3+2m(m-n)D(x^2)x^2,x\in R.$$

$$2(m+n{)}^{2}D(x^4)=2m(n-m)D(x)x^3+2m(3m+n)D(x)x^3+(m+n)(m-n)D(x^4),x\in R,$$

$$(m+n)(m+3n)D(x^4)=4m(m+n)D(x)x^3,x\in R.$$

$$(m+n{)}^2(m+3n)D(x^4)=4m(m+n{)}^{2}D(x)x^3,x\in R.$$

$$(m+n{)}^2(m+3n)D(x^4)=4m^2(m+3n)D(x)x^3,x\in R.$$

$$4mn(m-n)D(x)x^3=0,x\in R.$$

$$(m+n)D(x^2)=mD(x)x,x\in R.$$

$$(m+n)D(x^4)=mD(x^2)x^2,x\in R.$$

$$(m+n{)}^{2}D(x^4)=m^{2}D(x)x^3,x\in R.$$

$$2(m+n{)}^{2}D(x^4)=mnD(x)x^3+m(2m+n)D(x)x^3-mnD(x^2)x^2,x\in R.$$

$$2(m+n{)}^{2}D(x^4)=2m^{2}D(x)x^3,x\in R.$$

$$2(m+n{)}^{2}D(x^4)=mnD(x)x^3+m(2m+n)D(x)x^3-n(m+n)D(x^4),x\in R,$$

$$(m+n)(2m+3n)D(x^4)=2m(m+n)D(x)x^3,x\in R.$$

$$(m+n{)}^2(2m+3n)D(x^4)=2m(m+n{)}^{2}D(x)x^3,x\in R.$$

$$(m+n{)}^2(2m+3n)D(x^4)=m^2(2m+3n)D(x)x^3,x\in R.$$

$$mn(2n+m)D(x)x^3=0,x\in R.$$

Full text

On Generalized $(m,n)$-Jordan Derivations and Centralizers of Semiprime Rings

Driss Bennis${}^{1,\textbf{a}}$, Basudeb Dhara2 and Brahim Fahid${}^{1,\textbf{b}}$

1: Centre de Recherche de Mathématiques et Applications de Rabat (CeReMAR), Faculty of Sciences, Mohammed V University in Rabat, Rabat, Morocco

a: d.bennis@fsr.ac.ma; driss$\_$bennis@hotmail.com

b: fahid.brahim@yahoo.fr

2: Department of Mathematics, Belda College, Belda,

Paschim Medinipur, 721424, W.B., India.

basu$\_$dhara@yahoo.com

Abstract. In this paper we give an affirmative answer to two conjectures on generalized $(m,n)$-Jordan derivations and generalized $(m,n)$-Jordan centralizers raised in [S. Ali and A. Fošner, On Generalized $(m,n)$-Derivations and Generalized $(m,n)$-Jordan Derivations in Rings, Algebra Colloq. 21 (2014), 411–420] and [A. Fošner, A note on generalized $(m,n)$-Jordan centralizers, Demonstratio Math. 46 (2013), 254–262]. Precisely, when $R$ is a semiprime ring, we prove, under some suitable torsion restrictions, that every nonzero generalized $(m,n)$-Jordan derivation (resp., a generalized $(m,n)$-Jordan centralizer) is a derivation (resp., a two-sided centralizer).

2010 Mathematics Subject Classification. 16N60, 16W25

Key Words. semiprime ring, generalized $(m,n)$-derivation, generalized $(m,n)$-Jordan derivation, $(m,n)$-Jordan centralizer, generalized $(m,n)$-Jordan centralizer

1 Introduction

Throughout this paper, $R$ will represent an associative ring with center $Z(R)$. We denote by $char(R)$ the characteristic of a prime ring $R$. Let $n\geq 2$ be an integer. A ring $R$ is said to be $n$-torsion free if, for all $x\in R$, $nx=0$ implies $x=0$. Recall that a ring $R$ is prime if, for any $a,b\in R$, $aRb=\{0\}$ implies $a=0$ or $b=0$. A ring $R$ is called semiprime if, for any $a\in R$, $aRa=\{0\}$ implies $a=0$.

An additive mapping $d:R\longrightarrow R$ is called a derivation, if $d(xy)=d(x)y+xd(y)$ holds for all $x,y\in R$, and it is called a Jordan derivation, if $d(x^{2})=d(x)x+xd(x)$ holds for all $x\in R$. An additive mapping $T:R\longrightarrow R$ is called a left (resp., right) centralizer if $T(xy)=T(x)y$ (resp., $T(xy)=xT(y)$) is fulfilled for all $x,y\in R$, and it is called a left (resp., right) Jordan centralizer if $T(x^{2})=T(x)x$ (resp., $T(x^{2})=xT(x)$) is fulfilled for all $x\in R$. We call an additive mapping $T:R\longrightarrow R$ a two-sided centralizer (resp., a two-sided Jordan centralizer) if $T$ is both a left as well as a right centralizer (resp., a left and a right Jordan centralizer).

An additive mapping $F:R\longrightarrow R$ is called a generalized derivation if $F(xy)=F(x)y+xd(y)$ holds for all $x,y\in R$, where $d:R\longrightarrow R$ is a derivation. The concept of generalized derivations was introduced by Brešar in [3] and covers both the concepts of derivations and left centralizers. It is easy to see that generalized derivations are exactly those additive mappings $F$ which can be written in the form $F=d+T$, where $d$ is a derivation and $T$ is a left centralizer.

The Jordan counterpart of the notion of generalized derivation was introduced by Jing and Lu in [10] as follows: An additive mapping $F:R\longrightarrow R$ is called a generalized Jordan derivation if $F(x^{2})=F(x)x+xd(x)$ is fulfilled for all $x\in R$, where $d:R\longrightarrow R$ is a Jordan derivation.

The study of relations between various sorts of derivations goes back to Herstein’s classical result [9] which shows that any Jordan derivation on a $2$-torsion free prime ring is a derivation (see also [5] for a brief proof of Herstein’s result). In [7], Cusack generalized Herstein’s result to $2$-torsion free semiprime rings (see also [2] for an alternative proof). Motivated by these classical results, Vukman [17] proved that any generalized Jordan derivation on a $2$-torsion free semiprime ring is a generalized derivation.

In the last few years several authors have introduced and studied various sorts of parameterized derivations. In [1], Ali and Fošner defined the notion of $(m,n)$-derivations as follows: Let $m,n\geq 0$ be two fixed integers with $m+n\neq 0$. An additive mapping $d:R\longrightarrow R$ is called an $(m,n)$-derivation if

[TABLE]

holds for all $x,y\in R$.

Obviously, a $(1,1)$-derivation on a $2$-torsion free ring is a derivation.

In the same paper [1], a generalized $(m,n)$-derivation was defined as follows: Let $m,n\geq 0$ be two fixed integers with $m+n\neq 0$. An additive mapping $D:R\longrightarrow R$ is called a generalized $(m,n)$-derivation if there exists an $(m,n)$-derivation $d:R\longrightarrow R$ such that

[TABLE]

holds for all $x,y\in R$.

Obviously, every generalized $(1,1)$-derivation on a $2$-torsion free ring is a generalized derivation.

In [18], Vukman defined an $(m,n)$-Jordan derivation as follows: Let $m,n\geq 0$ be two fixed integers with $m+n\neq 0$. An additive mapping $d:R\longrightarrow R$ is called an $(m,n)$-Jordan derivation if

[TABLE]

holds for all $x,y\in R$.

Clearly, every $(1,1)$-Jordan derivation on a $2$-torsion free ring is a Jordan derivation.

Recently, in [11], Kosi-Ulbl and Vukman proved the following result.

Theorem 1.1** **([11], Theorem 1.5)

Let $m,n\geq 1$ be distinct integers, $R$ a $mn(m+n)|m-n|$-torsion free semiprime ring an $d:R\longrightarrow R$ an $(m,n)$-Jordan derivation. Then $d$ is a derivation which maps $R$ into $Z(R)$.

The $(m,n)$-generalized counterpart of the notion of an $(m,n)$-Jordan derivation is introduced by Ali and Fošner in [1] as follows: Let $m,n\geq 0$ be two fixed integers with $m+n\neq 0$. An additive mapping $F:R\longrightarrow R$ is called a generalized $(m,n)$-Jordan derivation if there exists an $(m,n)$-Jordan derivation $d:R\longrightarrow R$ such that

[TABLE]

holds for all $x,y\in R$.

Based on some observations and inspired by the classical results, Ali and Fošner in [1] made the following conjecture.

Conjecture 1.2** **([1], Conjecture 1)

Let $m,n\geq 1$ be two fixed integers, let $R$ be a semiprime ring with suitable torsion restrictions, and let $F:R\longrightarrow R$ be a nonzero generalized $(m,n)$-Jordan derivation. Then $F$ is a derivation which maps $R$ into $Z(R)$.

The first aim of this paper is to give an affirmative answer to this conjecture. Namely, our first main result is the following theorem.

Theorem 1.3

Let $m,n\geq 1$ be distinct integers, let $R$ be a $k$-torsion free semiprime ring, where $k=6mn(m+n)|m-n|$, and let $F:R\longrightarrow R$ be a nonzero generalized $(m,n)$-Jordan derivation. Then $F$ is a derivation which maps $R$ into $Z(R)$.

On the other hand and in parallel, there are similar works which study relations between various sorts of Jordan centralizers and centralizers. Namely, in [20], Zalar proved that any left (resp., right) Jordan centralizer on a $2$-torsion free semiprime ring is a left (resp., right) centralizer. In [15], Vukman proved that, for a $2$-torsion free semiprime ring $R$, every additive mapping $T:R\longrightarrow R$ satisfying the relation “$2T(x^{2})=T(x)x+xT(x)$ for all $x\in R$” is a two-sided centralizer. Motivated by these results and inspired by his work [15], Vukman in [19] introduced the notion of an $(m,n)$-Jordan centralizer as follows: Let $m,n\geq 0$ be two fixed integers with $m+n\neq 0$. An additive mapping $T:R\longrightarrow R$ is called an $(m,n)$-Jordan centralizer if

[TABLE]

holds for all $x,y\in R$.

Obviously, a $(1,0)$-Jordan centralizer (resp., $(0,1)$-Jordan centralizer) is a left (resp., a right) Jordan centralizer. When $n=m=1$, we recover the maps studied in [15].

Based on some observations and results, Vukman conjectured that, on semiprime rings with suitable torsion restrictions, every $(m,n)$-Jordan centralizer is a two-sided centralizer (see [19]). Recently, this conjecture was solved affirmatively by Kosi-Ulbl and Vukman in [12]. Namely, they proved the following result.

Theorem 1.4** **([12], Theorem 1.5)

Let $m,n\geq 1$ be distinct integers, let $R$ be an $mn(m+n)$-torsion free semiprime ring, and let $T:R\longrightarrow R$ be an $(m,n)$-Jordan centralizer. Then $T$ is a two-sided centralizer.

Inspired by the work of Vukman [15, 19], Fošner [8] introduced more generalized version of $(m,n)$-Jordan centralizers as follows: Let $m,n\geq 0$ be two fixed integers with $m+n\neq 0$. An additive mapping $T:R\longrightarrow R$ is called a generalized $(m,n)$-Jordan centralizer if there exists an $(m,n)$-Jordan centralizer $T_{0}:R\longrightarrow R$ such that

[TABLE]

holds for all $x\in R$.

Thus, a generalized $(1,0)$-Jordan centralizer is a left Jordan centralizer.

In [8], Fošner showed that, on a prime ring with a specific torsion condition, every generalized $(m,n)$-Jordan centralizer is a two-sided centralizer. This led Fošner to make the following conjecture.

Conjecture 1.5** **([8], Conjecture 1)

Let $m,n\geq 1$ be two fixed integers, let $R$ be a semiprime ring with suitable torsion restrictions, and let $T:R\longrightarrow R$ be a generalized $(m,n)$-Jordan centralizer. Then $T$ is a two-sided centralizer.

The second aim of this paper is to give an affirmative answer to Fošner’s conjecture. Namely, our second main result is the following theorem.

Theorem 1.6

Let $m,n\geq 1$ be two fixed integers, let $R$ be an $6mn(m+n)(2n+m)$-torsion free semiprime ring, and let $T:R\longrightarrow R$ be a nonzero generalized $(m,n)$-Jordan centralizer. Then $T$ is a two-sided centralizer.

2 Proof of the main theorems

In the proof of our main results, Theorems 1.3 and 1.6, we shall use the following results.

Lemma 2.1** **([1], Lemma 1)

Let $m,n\geq 0$ be distinct integers with $m+n\neq 0$, let $R$ be a $2$-torsion free ring, and let $F:R\longrightarrow R$ be a nonzero generalized $(m,n)$-Jordan derivation with an associated $(m,n)$-Jordan derivation $d$. Then, $(m+n)^{2}F(xyx)=m(n-m)F(x)xy+m(3m+n)F(x)yx+m(m-n)F(y)x^{2}+4mnxd(y)x+n(n-m)x^{2}d(y)+n(m+3n)xyd(x)+n(m-n)yxd(x)$ for all $x,y\in R$.

Lemma 2.2** **([6], Theorem 3.3)

Let $n\geq 2$ be a fixed integer and let $R$ be a prime ring with $char(R)=0$ or $char(R)\geq n$. If $T:R\longrightarrow R$ is an additive mapping satisfying the relation $T(x^{n})=T(x)x^{n-1}$ for all $x\in R$, then $T(xy)=T(x)y$ for all $x,y\in R$.

Lemma 2.3** **([8], Lemma 1)

Let $m,n\geq 0$ be distinct integers with $m+n\neq 0$, let $R$ be a ring, and let $T:R\longrightarrow R$ be a nonzero generalized $(m,n)$-Jordan centralizer with an associated $(m,n)$-Jordan centralizer $T_{0}$. Then, $2(m+n)^{2}T(xyx)=mnT(x)xy+m(2m+n)T(x)yx-mnT(y)x^{2}+2mnxT_{0}(y)x-mnx^{2}T_{0}(y)+n(m+2n)xyT_{0}(x)+mnyxT_{0}(x)$ for all $x,y\in R$.

Lemma 2.4** **([16], Lemma 3)

Let $R$ be a semiprime ring and let $T:R\longrightarrow R$ be an additive mapping. If either $T(x)x=0$ or $xT(x)=0$ holds for all $x\in R$, then $T=0$.

We shall use the relation between semiprime rings and prime ideals. Namely, it is well-known that a ring $R$ is semiprime if and only if the intersection of all prime ideals of $R$ is zero if and only if $R$ has no nonzero nilpotent (left, right) ideals (see for instance Lam’s book [13] or the recent book of Brešar [4]). Due to the classical Levitzki’s paper [14], several authors prefer to refer to a such result by Levitzki’s lemma.

Let $I$ be an ideal of $R$. For an element $x\in R$, we use $\overline{x}$ to denote the equivalence class of $x$ modulo $I$.

Lemma 2.5

Let $R$ be both a $2$-torsion free and a $3$-torsion free semiprime ring and let $T:R\longrightarrow R$ be an additive map such that $T(x)x^{3}=0$ and $T(x^{4})=0$ for all $x\in R$. Then $T(xy)=T(x)y$ for all $x,y\in R$.

**Proof. **Let $x,y\in R$. We prove that $T(xy)=T(x)y$. We may assume that $x$ and $y$ are not [math]. Let $P$ be a prime ideal of $R$ and set $\overline{R}=R/P$. Consider an element $p\in P$. By hypothesis, $0=T(x+p)(x+p)^{3}=(T(x)+T(p))(x^{3}+xpx+px^{2}+p^{2}x+x^{2}p+xp^{2}+pxp+p^{3})$. Thus, $0=T(x)(xpx+px^{2}+p^{2}x+x^{2}p+xp^{2}+pxp+p^{3})+T(p)(x^{3}+xpx+px^{2}+p^{2}x+x^{2}p+xp^{2}+pxp)$. Hence, $T(p)x^{3}\in P$, equivalently $\overline{T(p)}\overline{x}^{3}=0$. By Levitzki’s lemma, $\overline{T(p)}\overline{x}=0$, and then $\overline{T(p)}=0$ (since $\overline{R}$ is a prime ring). Thus, $T(P)\subseteq P$, which implies that $T(x+P)=T(x)+P$. Then, the induced map $\overline{T}:R/P\rightarrow R/P$ such that $\overline{T}(\overline{x})=\overline{T(x)}$ for every $x\in R$, is well defined. Now, since $\overline{T}(\overline{x})\overline{x}^{3}=0$ and $\overline{T}(\overline{x}^{4})=0$, $\overline{T}(\overline{x}^{4})=\overline{T}(\overline{x})\overline{x}^{3}$. This shows, using Lemma 2.2, that $\overline{T}(\overline{xy})=\overline{T}(\overline{x})\overline{y}$. Therefore, $T(xy)-T(x)y\in P$. Finally, by the semiprimeness of $R$, we get the desired result.

Now we are ready to prove the first main result.

Proof of Theorem 1.3. Let $d$ be the associated $(m,n)$-Jordan derivation of $F$. Since $R$ is a semiprime ring, $d$ is a derivation which maps $R$ into $Z(R)$ (by Theorem 1.1). Let us denote $F-d$ by $D$. Then, we have $(m+n)D(x^{2})=(m+n)F(x^{2})-(m+n)d(x^{2})=2mF(x)x+2nxd(x)-2md(x)x-2nxd(x)=2mD(x)x$ for all $x\in R$. Thus

[TABLE]

Replacing $x$ with $x^{2}$ in (2.1), we get

[TABLE]

Multiplying by $m+n$ and then using (2.1), we get

[TABLE]

On the other hand, putting $x^{2}$ for $y$ in the relation of Lemma 2.1 and using the fact that $D$ is a generalized $(m,n)$-Jordan derivation associated with the zero map as an $(m,n)$-Jordan derivation, we get

[TABLE]

Multiplying both sides in (2.4) by $2$ we get

[TABLE]

Combining (2.2) and (2.5), we get

[TABLE]

which gives

[TABLE]

Multiplying both sides in (2.7) by $m+n$, we get

[TABLE]

Multiplying by $m+3n$ in (2.3), we get

[TABLE]

By comparing (2.8) and (2.9), we get

[TABLE]

Since $R$ is a $2mn|n-m|$-torsion free ring, $D(x)x^{3}=0$ for all $x\in R$. Applying $D(x)x^{3}=0$ in equation (2.3), we get $(m+n)^{2}D(x^{4})=0$ for all $x\in R$. By using the torsion free restriction, we have $D(x^{4})=0$ for all $x\in R$. Hence, $D(xy)=D(x)y$ for all $x,y\in R$ (by Lemma 2.5). Applying this in (2.1), yields $(m+n)D(x)x=2mD(x)x$ for all $x\in R$, equivalently $(m-n)D(x)x=0$. Since $R$ is an $|m-n|$-torsion free ring, $D(x)x=0$ for all $x\in R$. Therefore, by Lemma 2.4, $D=0$. This completes the proof.

The second main result is proved similarly. Nevertheless, we include a proof for completeness.

Proof of Theorem 1.6. Let $T_{0}$ be the associated $(m,n)$-Jordan centralizer of $T$. Since $R$ is a semiprime ring, $T_{0}$ is a two-sided centralizer (by Theorem 1.4). Let us denote $T-T_{0}$ by $D$. Then, we have $(m+n)D(x^{2})=(m+n)T(x^{2})-(m+n)T_{0}(x^{2})=mT(x)x+nxT_{0}(x)-mT_{0}(x)x-nxT_{0}(x)=mD(x)x$ for all $x\in R.$ Thus

[TABLE]

Replacing $x$ with $x^{2}$ in (2.11), we get

[TABLE]

Multiplying by $m+n$ and then using (2.11), we get

[TABLE]

On the other hand, if we put $y=x^{2}$ in the relation of Lemma 2.3, we get

[TABLE]

Multiplying both sides in (2.13) by $2$ we get

[TABLE]

Combining (2.12) and (2.14), we get

[TABLE]

which implies

[TABLE]

Multiplying both sides of above relation by $m+n$, we have

[TABLE]

Multiplying by $(2m+3n)$ in (2.13), we get

[TABLE]

By comparing (2.18) and (2.19), we get

[TABLE]

Since $R$ is a $mn(2n+m)$-torsion free ring, $D(x)x^{3}=0$ for all $x\in R$. Applying $D(x)x^{3}=0$ in equation (2.13) and then using $(m+n)$-torsion freeness of $R$, we get $D(x^{4})=0$ for all $x\in R$. Moreover, since $R$ is a $2$ and a $3$-torsion free ring, by Lemma 2.5, we get $D(xy)=D(x)y$ for all $x,y\in R$. Applying this in (2.11), yields $(m+n)D(x)x=mD(x)x$ for all $x\in R$. So $nD(x)x=0$, which implies that $D(x)x=0$ for all $x\in R$. Therefore, by Lemma 2.4, $D=0$. This completes the proof.

Acknowledgements. The authors would like to thank Professor Abdellah Mamouni for useful discussions.