\def\CTeXPreproc{Created by ctex v0.2.12, don't edit!}\documentstyle[12pt]{article} \topmargin=10mm \textheight=200mm \textwidth=150mm %\leftmargin=-5mm %\input{input} \begin{document} \thispagestyle{empty} \sloppy %\baselineskip=8mm \begin{center} {\bf{\large{ Generalized derivations in prime and semiprime rings$^*$ }}} \vspace{.5cm} \begin{center} Shuliang Huang and Nadeem ur Rehman \vspace{.5cm} \end{center} \begin{abstract} Let $R$ be a prime ring, $I$ a nonzero ideal of $R$ and $m, n$ fixed positive integers. If $R$ admits a generalized derivation $F$ associated with a nonzero derivation $d$ such that $(F([x,y])^{m}=[x,y]_{n}$ for all $x,y\in I$, then $R$ is commutative. Moreover we also examine the case when $R$ is a semiprime ring. \end{abstract} \footnote{$^*$Supported by the Natural Science Research Foundation of Anhui Provincial Education Department(No. KJ2010B144) of P. R. CHINA.} \\ \end{center} \noindent{\bf Keywords:} prime and semiprime rings, generalized derivations, GPIs. \noindent{\bf 2000 Mathematics Subject Classification}: 16N60, 16U80, 16W25. \vspace{.5cm} \parindent=0mm {\bf{1. Introduction}} \\ \parindent=8mm In all that follows, unless stated otherwise, $R$ will be an associative ring, $Z(R)$ the center of $R$, $Q$ its Martindale quotient ring and $U$ its Utumi quotient ring. The center of $U$, denoted by $C$, is called the extended centroid of $R$ (we refer the reader to [3] for these objects). For any $x, y \in R$, the symbol $[x,y]$ and $x\circ y $ stand for the commutator $xy-yx$ and anti-commutator $xy+yx$, respectively. For each $x, y \in R$ and each $n\geq 1$, define $[x,y]_{1}=xy-yx$ and $[x,y]_{k}=[[x,y]_{k-1},y]$ for $k\geq 2$. Recall that a ring $R$ is prime if for any $a, b\in R$, $aRb=(0)$ implies $a=0$ or $b=0$, and is 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 $x,y\in R$. In [4], Bresar introduced the definition of generalized derivation: an additive mapping $F:R \longrightarrow R$ is called a generalized derivation if there exists a derivation $d:R \longrightarrow R$ such that $F(xy)=F(x)y+xd(y)$ holds for all $x,y\in R$, and $d$ is called the associated derivation of $F$. Hence, the concept of generalized derivations covers both the concepts of a derivation and of a left multiplier. Basic examples are derivations and generalized inner derivations. We refer to call such mappings generalized inner derivations for the reason they present a generalization of the concept of inner derivations. In [9], Hvala studied generalized derivations in the context of algebras on certain norm spaces. In [13], Lee extended the definition of a generalized derivation as follows: by a generalized derivation we mean an additive mapping $F:I \longrightarrow U$ such that $F(xy)=F(x)y+xd(y)$ holds for all $x,y\in I$, where $I$ is a dense left ideal of $R$ and $d$ is a derivation from $I$ into $U$. Moreover, Lee also proved that every generalized derivation can be uniquely extended to a generalized derivation of $U$ and thus all generalized derivations of $R$ will be implicitly assumed to be defined on the whole of $U$. Lee obtained the following: every generalized derivation $F$ on a dense left ideal of $R$ can be uniquely extended to $U$ and assumes the form $F(x)=ax+d(x)$ for some $a\in U$ and a derivation $d$ on $U$.\\ \parindent=8mm This paper is included in a line of investigation concerning the relationship between the structure of a ring $R$ and the behaviour of some additive mappings defined on $R$ satisfy certain special identities. In [1], Ashraf and Rehman proved that if $R$ is a prime ring, $I$ a nonzero ideal of $R$ and $d$ is a derivation of $R$ such that $d(x\circ y )=x\circ y$ for all $x,y\in I$, then $R$ is commutative. In [2, Theorem 1], Argac and Inceboz generalized the above result as following: Let $R$ be a prime ring, $I$ a nonzero ideal of $R$ and $n$ a fixed positive integer, if $R$ admits a derivation $d$ with the property $(d(x\circ y ))^{n}=x\circ y$ for all $x,y\in I$, then $R$ is commutative. In [7], Daif and Bell showed that if in a semiprime ring $R$ there exists a nonzero ideal $I$ of $R$ and a derivation $d$ such that $d([x,y])=[x,y]$ for all $x,y\in I$, then $I\subseteq Z(R)$. At this point the natural question is what happens in case the derivation is replaced by a generalized derivation. In [18], Quadri et al., proved that if $R$ is a prime ring, $I$ a nonzero ideal of $R$ and $F$ a generalized derivation associated with a nonzero derivation $d$ such that $F([x,y])=[x,y]$ for all $x,y\in I$, then $R$ is commutative. In [10], we studied a similar condition and proved that a prime ring $R$ satisfying $(F(x\circ y))^{n}=x\circ y$ must be commutative. The present paper is motivated by the previous results and we here continue this line of investigation by examining what happens a ring $R$ satisfying the identity $(F([x,y])^{m}=[x,y]_{n}$. Explicitly we shall prove the following:\\ \parindent=0mm {\bf{Theorem 1}} Let $R$ be a prime ring, $I$ a nonzero ideal of $R$ and $m, n$ fixed positive integers. If $R$ admits a generalized derivation $F$ associated with a nonzero derivation $d$ such that $(F([x,y])^{m}=[x,y]_{n}$ for all $x,y\in I$, then $R$ is commutative. \\ {\bf{Theorem 2}} Let $R$ be a semiprime ring and $m, n$ fixed positive integers. If $R$ admits a generalized derivation $F$ associated with a derivation $d$ such that $(F([x,y])^{m}=[x,y]_{n}$ for all $x,y\in R$, then there exists a central idempotent element $e$ in $U$ such that on the direct sum decomposition $R=eU\oplus (1-e)U$, $d$ vanishes identically on $eU$ and the ring $(1-e)U$ is commutative.\\ \parindent=8mm \parindent=0mm %\newpage {\large{\bf{2. The case: $R$ a prime ring }}} \\ \parindent=0mm {\bf{Theorem 2.1.}} Let $R$ be a prime ring, $I$ a nonzero ideal of $R$ and $m, n$ fixed positive integers. If $R$ admits a generalized derivation $F$ associated with a nonzero derivation $d$ such that $(F([x,y])^{m}=[x,y]_{n}$ for all $x,y\in I$, then $R$ is commutative. \\ \parindent=0mm {\bf{Proof.}}~Since $R$ is a prime ring and $F$ is a generalized derivation of $R$, by Lee [13, Theorem 3], $F(x)=ax+d(x)$ for some $a\in U$ and a derivation $d$ on $U$. By the given hypothesis we have now $[x,y]_{n}=(a[x,y]+d([x,y]))^{m}=(a[x,y]+[d(x),y]+[x,d(y)])^{m}$ for all $x,y\in I$. By Kharchenko [12], we divide the proof into two cases:\\ Case 1. Let $d$ be an outer derivation of $U$, then $I$ satisfies the polynomial identity $(a[x,y]+[s,y]+[x,t])^{m}=[x,y]_{n}$ for all $x,y,s,t\in I$. In particular, for $y=0$, $I$ satisfies the blended component $([x,t])^{m}=0$ for all $x,t\in I$, by Herstein [11, Theorem 2], we have $I\subseteq Z(R)$, and so $R$ is commutative by Mayne [17, Lemma 3].\\ Case 2. Let now $d$ be the inner derivation induced by an element $q\in Q$, that is $d(x)=[q,x]$ for all $x,y\in U$. It follows that $(a[x,y]+[[q,x],y]+[x,[q,y]])^{m}=[x,y]_{n}$ for all $x,y\in I$. By Chuang [5, Theorem 2], $I$ and $Q$ satisfy the same generalized polynomial identities (GPIs), we have $(a[x,y]+[[q,x],y]+[x,[q,y]])^{m}=[x,y]_{n}$ for all $x,y\in Q$. In case center $C$ of $Q$ is infinite, we have $(a[x,y]+[[q,x],y]+[x,[q,y]])^{m}=[x,y]_{n}$ for all $x,y\in Q\bigotimes_{C} \overline{C}$, where $\overline{C}$ is the algebraic closure of $C$. Since both $Q$ and $Q\bigotimes_{C} \overline{C}$ are prime and centrally closed [8, Theorem 2.5 and Theorem 3.5], we may replace $R$ by $Q$ or $Q\bigotimes_{C} \overline{C}$ according as $C$ is finite or infinite. Thus we may assume that $R$ is centrally closed over $C$ (i.e. $RC=C$) which is either finite or algebraically closed and $(a[x,y]+[[q,x],y]+[x,[q,y]])^{m}=[x,y]_{n}$ for all $x,y\in R$. By Martindale [16, Theorem 3], $RC$ (and so $R$) is a primitive ring which is isomorphic to a dense ring of linear transformations of a vector space $V$ over a division ring $D$.\\ Assume that $dim V_{D}\geq 3$. First of all, we want to show that $v$ and $qv$ are linearly $D$-dependent for all $v\in V$. Since if $qv=0$ then ${v, qv}$ is $D$-dependent, suppose that $qv\neq0$. If $v$ and $qv$ are $D$-independent, since $dim V_{D}\geq 3$, then there exists $w\in V$ such that $v, qv, w$ are also $D$-independent. By the density of $R$, there exists $x,y\in R$ such that: $xv=0, xqv=w, xw=v; yv=0, yqv=0, yw=v$. These imply that $v=(a[x,y]+[[q,x],y]+[x,[q,y]])^{m}v=[x,y]_{n}v=0$, which is a contradiction. So we conclude that $v$ and $qv$ are linearly $D$-dependent for all $v\in V$. Our next goal is to show that there exists $b\in D$ such that $qv=vb$ for all $v\in V$. In fact, choose $v,w\in V$ linearly independent. Since $dim V_{D}\geq 3$, then there exists $u\in V$ such that $u,v,w$ are linearly independent, and so $b_{u}, b_{v}, b_{w} \in D$ such that $qu=ub_{u}$, $qv=vb_{v}$, $qw=wb_{w}$, that is $q(u+v+w)=ub_{u}+vb_{v}+wb_{w}$. Moreover $q(u+v+w)=(u+v+w)b_{u+v+w}$ for a suitable $b_{u+v+w}\in D$. Then $0=u(b_{u+v+w}-b_{u})+v(b_{u+v+w}-b_{v})+w(b_{u+v+w}-b_{w})$ and because $u,v,w$ are linearly independent, $b_{u}=b_{v}=b_{w}=b_{u+v+w}$, that is $b$ does not depend on the choice of $v$. Hence now we have $qv=vb$ for all $v\in V$. Now for $r\in R$, $v\in V$, we have $(rq)v=r(qv)=r(vb)=(rv)b=q(rv)$, that is $[q,R]V=0$. Since $V$ is a left faithful irreducible $R$-module, hence $[q,R]=0$, i.e. $q\in Z(R)$ and so $d=0$, a contradiction.\\ Suppose now that $dim V_{D}\leq2$. In this case $R$ is a simple GPI-ring with 1, and so it is a central simple algebra finite dimensional over its center. By Lanski [14, Lemma 2], it follows that there exists a suitable filed $F$ such that $R\subseteq M_{k}(F)$, the ring of all $k \times k$ matrices over $F$, and moreover $M_{k}(F)$ satisfies the same GPI as $R$. Assume $k\geq3$, by the same argument as in the above, we can get a contradiction. Obviously if $k=1$, then $R$ is commutative. Thus we may assume that $k=2$ i.e., $R\subseteq M_{2}(F)$, where $M_{2}(F)$ satisfies $(a[x,y]+[[q,x],y]+[x,[q,y]])^{m}=[x,y]_{n}$. Denote $e_{ij}$ the usual matrix unit with 1 in $(i,j)$-entry and zero elsewhere. Let $[x,y]=[e_{21},e_{11}]=e_{21}$. Then $[x,y]_{n}=e_{21}$. In this case we have $(ae_{21}+qe_{21}-e_{21}q)^{m}=e_{21}$. Right multiplying by $e_{21}$, we get $(-1)^{m}(e_{21}q)^{m}e_{21}=(ae_{21}+qe_{21}-e_{21}q)^{m}e_{21}=e_{21}e_{21}=0$. Set $q=\left(\begin{array}{cc} q_{11} & q_{12}\\ q_{21} & q_{22}\end{array}\right)$. By calculation we find that $(-1)^{m}\left(\begin{array}{cc} 0 & 0\\ q_{12}^{m} & 0\end{array}\right)=0$, which implies that $q_{12}=0$. Similarly we can see that $q_{21}=0$. Therefore $q$ is diagonal in $M_{2}(F)$. Let $f \in Aut(M_{2}(F))$. Since $(f(a)[f(x),f(y)]+[[f(q),f(x)],f(y)]+[f(x),[f(q),f(y)]])^{m}=[f(x),f(y)]_{n}$ so $f(q)$ must be a diagonal matrix in $M_{2}(F)$. In particular, let $f(x)=(1-e_{ij})x(1+e_{ij})$ for $i\neq j$, then $f(q)=q+(q_{ii}-q_{jj})e_{ij}$, that is $q_{ii}=q_{jj}$ for $i\neq j$. This implies that $q$ is central in $M_{2}(F)$, which leads to $d=0$, a contradiction. This completes the proof of the theorem.\\ The following example demonstrates that $R$ to be prime is essential in the hypothesis.\\ \parindent=0mm {\bf{Example 2.1.}} Consider $S$ be any ring and let $ R=\left\{\left(\begin{array}{cc} a & b\\ 0 & 0\end{array}\right)\mid a,b\in S\right\}$ and let $ I=\left \{\left(\begin{array}{cc} 0 & a\\ 0 & 0\end{array}\right)~\mid~ a\in S\right\}$ be a nonzero ideal of $R$. We define a map $F : R\rightarrow R$ by $F(x)=2e_{11}x-xe_{11}$. Then it is easy to see that $F$ is a generalized derivation associated with a nonzero derivation $d(x)=[e_{11},x]$. It is straightforward to check that $F$ satisfies the property: $(F([x,y])^{m}=[x,y]_{n}$ for all $x,y\in I$. However, $R$ is not commutative.\\ %\newpage {\large{\bf{3. The case: $R$ a semiprime ring }}} \\ \parindent=8mm \parindent=0mm {\bf{Theorem 3.1.}} Let $R$ be a semiprime ring and $m, n$ fixed positive integers. If $R$ admits a generalized derivation $F$ associated with a derivation $d$ such that $(F([x,y])^{m}=[x,y]_{n}$ for all $x,y\in R$, then there exists a central idempotent element $e$ in $U$ such that on the direct sum decomposition $R=eU\oplus (1-e)U$, $d$ vanishes identically on $eU$ and the ring $(1-e)U$ is commutative.\\ \parindent=0mm {\bf{Proof.}}~~Since $R$ is semiprime and $F$ is a generalized derivation of $R$, by Lee [13, Theorem 3], $F(x)=ax+d(x)$ for some $a\in U$ and a derivation $d$ on $U$. We are given that $(a[x,y]+d([x,y]))^{m}=[x,y]_{n}$ for all $x,y\in R$. By Lee [15,Theorem 3], $R$ and $U$ satisfy the same differential identities, then $(a[x,y]+d([x,y]))^{m}=[x,y]_{n}$ for all $x,y\in U$. Let $B$ be the complete Boolean algebra of idempotents in $C$ and $M$ be any maximal ideal of $B$. Since $U$ is a $B$-algebra orthogonal complete [6, p.42] and $MU$ is a prime ideal of $U$, which is $d$-invariant. Denote $\overline{U}=U/MU$ and $\overline{d}$ the derivation induced by $d$ on $\overline{U}$, i.e., $\overline{d}(\overline{u})=\overline{d(u)}$ for all $u\in U$. For all $\overline{x}, \overline{y}\in \overline{U}$, $(\overline{a}[\overline{x},\overline{y}]+\overline{d}([\overline{x},\overline{y}]))^{m}=[\overline{x},\overline{y}]_{n}$. It is obvious that $\overline{U}$ is prime. Therefore by Theorem 2.1, we have either $\overline{U}$ is commutative or $\overline{d}=0$, that is either $d(U) \subseteq MU$ or $[U,U]\subset MU$. Hence $d(U)[U,U]\subseteq MU$, where $MU$ runs over all prime ideals of $U$. Since $\cap_{M}MU=0$, we obtain $d(U)[U,U]=0$. \parindent=8mm By using the theory of orthogonal completion for semiprime rings( see [3, Chapter 3]), it is clear that there exists a central idempotent element $e$ in $U$ such that on the direct sum decomposition $R=eU\oplus (1-e)U$, $d$ vanishes identically on $eU$ and the ring $(1-e)U$ is commutative. This completes the proof of the theorem.\\ \parindent=0mm %\begin{center} \begin{thebibliography}{99} \bibitem{} M. Ashraf and N. Rehman, On commutativity of rings with derivations, Results Math., 42 (2002), no.1-2, 3-8. \bibitem{} N. Argac and H. G. Inceboz, Derivations of prime and semiprime rings, J. Korean Math. Soc., 46(2009), no.5, 997-1005. \bibitem {} K. I. Beidar, W. S. Martindale and V. Mikhalev, Rings with generalized identities, Monographs and Textbooks in Pure and Applied Mathematics, 196. Marcel Dekker, Inc., New York, 1996. \bibitem {} M. 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CHINA\\ E-mail:shulianghuang@163.com \vspace{.2cm} Nadeem ur Rehman\\ Department of Mathematics\\ Aligarh Muslim University\\ Aligarh 202002, INDIA\\ E-mail: rehman100@gmail.com\\ \end{document}