A NOTE ON A PAIR OF DERIVATIONS OF SEMIPRIME RINGS

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PII. S0161171204302139 http://ijmms.hindawi.com

A NOTE ON A PAIR OF DERIVATIONS OF SEMIPRIME RINGS

MUHAMMAD ANWAR CHAUDHRY and A. B. THAHEEM Received 17 February 2003

We study certain properties of derivations on semiprime rings. The main purpose is to prove the following result: let R be a semiprime ring with centerZ(R), and letf,gbe derivations ofRsuch thatf (x)x+xg(x)∈Z(R)for allx∈R, thenf andgare central.

As an application, we show that noncommutative semisimple Banach algebras do not admit nonzero linear derivations satisfying the above central property. We also show that every skew-centralizing derivationfof a semiprime ringRis skew-commuting.

2000 Mathematics Subject Classification: 47A50, 47B50.

1. Introduction and preliminaries. Throughout,Rdenotes a ring with centerZ(R).

We write[x, y]forxy−yx. We will frequently use the identities[xy, z]=x[y, z]+ [x, z]yand[x, yz]=y[x, z]+[x, y]zfor allx, y, z∈R. We recall thatRis semiprime ifaRa=(0)impliesa=0 and it is prime ifaRb=(0)impliesa=0 orb=0. A prime ring is semiprime but the converse is not true in general. An additive mappingd:R→R is called a derivation ifd(xy)=d(x)y+xd(y)for allx, y∈R. A mappingf:R→R is called centralizing if[f (x), x]∈Z(R)for allx∈R; in particular, if[f (x), x]=0 for all x∈R, then it is called commuting. A mapping f :R→R is called central if f (x)∈Z(R)for all x∈R. Every central mapping is obviously commuting but not conversely, in general. A lot of work has been done on centralizing mappings (see, e.g., [3,4,5] and the references therein). A mappingf :R→R is called skew-centralizing if f (x)x+xf (x)∈Z(R)for all x ∈R; in particular, if f (x)x+xf (x)=0 for all x∈R, then it is called skew-commuting. We denote the radical of a Banach algebraA by rad(A).

We now recall some facts concerning semiprime rings and their extended centroids.

For any semiprime ringR, one can construct the ring of quotientsQofR [1]. AsR can be embedded isomorphically inQ, we considerRas a subring ofQ. If the element q∈Qcommutes with every element in R, thenq belongs toC, the center of Q. C contains the centroid ofR and is called the extended centroid ofR. In general,Cis a von Neumann regular ring, and it is a field if and only ifRis a prime [1, Theorem 5].

For more information on extended centroid ofR, we refer to [2].

Brešar [6, Theorem 2] has proved that ifRis a prime ring of characteristic not 2 and f:R→Ris an additive skew-commuting mapping (i.e.,f satisfiesf (x)x+xf (x)=0 for allx∈R), thenf=0.

Moreover, Brešar [5, Theorem 4.1] has considered a pair of derivations on a prime ring and has proved the following. LetRbe a prime ring andUa nonzero left ideal ofR.

Suppose that the derivationsdandgofRare such thatd(u)u−ug(u)∈Z(R)for all u∈U. Ifd≠0, thenRis commutative.

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A mappingh:R→Rdefined byh(x)=ax+xb(x∈R) for somea, b∈Ris called a generalized inner derivation [8]. Generalized inner derivations are called elementary operators and have been extensively studied in operator algebras. We note that the condition thathis centralizing onRcan be written in the form[a, x]x+x[b, x]∈Z(R) for allx∈R. Thus, introducing inner derivationsfandgbyf (x)=[a, x]andg(x)= [b, x], we obtain the condition as in [5, Theorem 4.1], that is,f (x)x+xg(x)∈Z(R) for allx∈R.

Recently, Thaheem [9] has proved the following result.

Theorem1.1. Iff,gis a pair of derivations on a semiprime ringRsatisfyingf (x)x+ xg(x)=0for allx∈R, thenf (x), g(x)∈Z(R)andf (u)[x, y]=g(u)[x, y]=0for allu, x, y∈R.

Inspired by the works of Brešar [5, 6] and Thaheem [9] and the above remarks regarding generalized inner derivations, we consider a general situation regarding a pair of derivations of a semiprime ring and prove the following. Letf,gbe a pair of derivations of a semiprime ringRsatisfyingf (x)x+xg(x)∈Z(R)for allx∈R, thenf and gare central (Theorem 2.2). We also show that every skew-centralizing derivationfof a semiprime ringRis skew-commuting (Corollary 2.3).

We will need the following result of Brešar [7, Theorem 3.1] in the sequel.

Theorem 1.2. LetS be a set andR a semiprime ring. If functionsf and g of S intoRsatisfyf (s)xg(t)=g(s)xf (t)for alls, t∈S,x∈R, then there exist idempotents 1, 2, 3∈Cand an invertible elementλ∈Csuch thatij=0, fori≠j,1+2+3=1, and1f (s)=λ1g(s),2g(s)=0,3f (s)=0hold for alls∈S.

2. The results. We now prove our results.

Lemma2.1. Letf,gbe a pair of derivations of a semiprime ringRsatisfyingf (x)x+ xg(x)∈Z(R), thencf andcgare central for allc∈Z(R).

Proof. Ifc=0, then obviouslycf andcg are central. Letcbe a nonzero element ofZ(R). Linearizingf (x)x+xg(x)∈Z(R), we get

f (x)y+f (y)x+xg(y)+yg(x)∈Z(R) ∀x, y∈R. (2.1) Takingy=cin (2.1), we get

f (x)c+f (c)x+xg(c)+cg(x)∈Z(R) ∀x∈R. (2.2)

Replacingy=c²in (2.1), we obtain f (x)c²+2cf (c)x+x

2cg(c)

+c²g(x)∈Z(R), (2.3)

that is, c

f (x)c+cg(x)+f (c)x+xg(c) +c

f (c)x+xg(c)

∈Z(R). (2.4)

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Noting that the first summand is contained inZ(R)by (2.2), from (2.4), we obtain

c

f (c)x+xg(c)

∈Z(R) ∀x∈R. (2.5)

Thus

c

f (c)x+xg(c) , y

=0 ∀x, y∈R. (2.6)

This implies

c

f (c)x+xg(c), y

=0 ∀x, y∈R. (2.7)

Further, (2.2) implies

f (x)c+cg(x), y

= −

f (c)x+xg(c), y

∀x, y∈R. (2.8)

From (2.7) and (2.8), we obtainc[f (x)c+cg(x), y]=0, which implies

c²

f (x)+g(x), y

=0 ∀x, y∈R. (2.9)

Replacingybyzyin (2.9), we getc²z[f (x)+g(x), y]=0, which implies

czc

f (x)+g(x), y

=0 ∀x, y, z∈R. (2.10)

Replacingzby[f (x)+g(x), y]zin (2.10), we getc[f (x)+g(x), y]zc[f (x)+g(x), y]= 0, which, by semiprimeness of R, implies c[f (x)+g(x), y]=0; that is, [c(f (x)+ g(x)), y]=0 for allx, y∈R. Thus,

c

f (x)+g(x)

∈Z(R) ∀x∈R. (2.11)

Sincec∈Z(R)andf,gare derivations, thereforecf,cg, andc(f+g)are derivations of R. Further, (2.11) implies that c(f+g) is central and hence, by [3, Lemma 4], a commuting derivation. Thus, by Thaheem and Samman [10, Proposition 2.3], we get (c(f+g))(u)[x, y]=0 for allu, x, y∈R. That is,

c

f (u)+g(u)

[x, y]=0 ∀u, x, y∈R. (2.12)

Using (2.12) and the fact thatcf (u)+cg(u)∈Z(R), we get[(cf (u)+cg(u))u, y]= (cf (u)+cg(u))[u, y]+[cf (u)+cg(u), y]u=0; that is,

cf (u)u+cg(u)u, y

=0 ∀u, y∈R. (2.13)

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Sincec∈Z(R)andf (u)u+ug(u)∈Z(R), thereforecf (u)u+cug(u)∈Z(R). Thus cf (u)u+cug(u), y

=0 ∀u, y∈R. (2.14)

Subtracting (2.14) from (2.13), we get[cg(u)u−cug(u), y]=0; that is, [c(g(u)u− ug(u)), y] = [c[g(u), u], y]= [[cg(u), u], y] = 0 for all u, y ∈ R, which implies [cg(u), u]∈Z(R). Thus,cgis a centralizing derivation. By [3, Lemma 4], we get that cgis a commuting derivation. By Thaheem and Samman [10, Proposition 2.3], we get cg(u)∈Z(R). Thuscgis central.

Sincecf (u)+cg(u)∈Z(R)andcg(u)∈Z(R), thereforecf (u)∈Z(R). Socf is central.

Theorem2.2. LetR be a semiprime ring andf,ga pair of derivations ofR such thatf (x)x+xg(x)∈Z(R)for allx∈R. Thenf andgare central.

Proof. Let x0 ∈R and c= f (x0)x0+x0g(x0). Then, by hypothesis, c ∈Z(R).

ByLemma 2.1, cf and cg are central. Thus [cf (x), y]=0 for all x, y∈R. That is, cf (x)y−ycf (x)=0, which implies

f (x)yc=cyf (x) ∀x, y∈R. (2.15) TakingS =R, g(x)=c and applying Theorem 1.2 to (2.15), we get that there exist idempotents1, 2, 3∈Cand an invertible elementλ∈Csuch thatij=0 fori≠j, 1+2+3=1, and

1f (x)=λ1c, 2c=0, 3f (x)=0, ∀x∈R. (2.16) Replacing x by xy in the first identity of (2.16) and using it again, we get λ1c = 1f (xy)=1(f (x)y+xf (y))=1f (x)y+x1f (y)=λ1cy+xλ1c; that is,

λ1c=λ1cy+xλ1c ∀x, y∈R. (2.17) Replacingy by−xin (2.17), we getλ1c=λ1c(−x)+xλ1c= −xλ1c+xλ1c= 0.

Thus 1f (x)=λ1c=0 for all x∈R. Hence, using (2.16), we get f (x)=(1+2+ 3)f (x)=2f (x), which impliescf (x)=c2f (x)=2cf (x)=0. Thuscf (x)=0 for allx∈R. Sincecg is central, therefore, analogously, it follows thatcg(x)=0 for all x∈R. Hencecf (x)x=0 andcxg(x)=0 for allx∈R. Thusc(f (x)x+xg(x))=0. In particular, 0=c(f (x0)x0+x0g(x0))=c². Since a semiprime ring has no nonzero central nilpotents, thereforec=0; that is,f (x0)x0+x0g(x0)=0. Sincex0is an arbitrary element ofR, therefore

f (x)x+xg(x)=0 ∀x∈R. (2.18) UsingTheorem 1.1, from (2.18), we get thatfandgare central.

Taking g(x)= f (x) in Theorem 2.2 and considering (2.18), we get the following corollary.

Corollary2.3. Letfbe a skew-centralizing derivation of a semiprime ringR, then f is skew-commuting.

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Corollary2.4. LetRbe a noncommutative prime ring andf,ga pair of derivations ofRsuch thatf (x)+xg(x)∈Z(R)for allx∈R, thenf=g=0.

Proof. SinceR, being prime, is semiprime, therefore, by (2.18), we get f (x)x+ xg(x)=0 for allx∈R. ThenTheorem 1.1gives

f (u)[x, y]=0=g(u)[x, y] ∀u, x, y∈R. (2.19) Replacingybyzyin (2.19) and using (2.19) again, we getf (u)z[x, y]=0=g(u)z[x, y]. SinceR is prime and noncommutative, thereforef (u)=0=g(u)for allu∈R.

Thusf=g=0.

It is well known that there are no nonzero linear derivations on a commutative semisimple Banach algebra. Thus, it is natural to identify situations under which noncommutative semisimple Banach algebras do not admit nontrivial derivations. The following corollary, which follows as an application of our results, identifies such a situation.

Corollary2.5. LetAbe a noncommutative semisimple Banach algebra with center Z(A)and letf,gbe a pair of linear derivations ofAsuch thatf (x)x+xg(x)∈Z(A) for allx∈A. Thenf=g=0.

Proof. SinceAis semisimple, therefore it is semiprime. Thus, byTheorem 2.2,f andgare central and trivially commuting as well as centralizing. Hence, by [4, Corollary 3.7],f andgmapAintoZ(A)∩rad(A). SinceAis semisimple, therefore rad(A)=(0).

Thusf (x)=0=g(x)for allx∈A. Hencef=0 andg=0.

Remark2.6. (i) Takingg(x)=f (x)inCorollary 2.5, we get that noncommutative semisimple Banach algebras do not admit nontrivial linear skew-centralizing derivations.

(ii) Takingg(x)=f (x)inTheorem 2.2, we get that every skew-centralizing derivationfof a semiprime ringRis central.

Acknowledgments. The authors gratefully acknowledge the support provided by King Fahd University of Petroleum and Minerals during this research. The authors would like to thank the referees for their valuable comments that led to the improvement of this note.

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Muhammad Anwar Chaudhry: Department of Mathematical Sciences, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia

E-mail address:[email protected]

A. B. Thaheem: Department of Mathematical Sciences, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia

E-mail address:[email protected]