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Apr 19, 2011 - Shakir Ali. Abstract. The purpose of this paper is to establish some results concerning generalized left derivations in rings and Banach algebras ...
Aequat. Math. 81 (2011), 209–226 c Springer Basel AG 2011  0001-9054/11/030209-18 published online April 19, 2011 DOI 10.1007/s00010-011-0070-5

Aequationes Mathematicae

On generalized left derivations in rings and Banach algebras Shakir Ali

Abstract. The purpose of this paper is to establish some results concerning generalized left derivations in rings and Banach algebras. In fact, we prove the following results: Let R be a 2-torsion free semiprime ring, and let G : R −→ R be a generalized Jordan left derivation with associated Jordan left derivation δ : R −→ R. Then every generalized Jordan left derivation is a generalized left derivation on R. This result gives an affirmative answer to the question posed as a remark in Ashraf and Ali (Bull. Korean Math. Soc. 45:253–261, 2008). Also, the study of generalized left derivation has been made which acts as a homomorphism or as an anti-homomorphism on some appropriate subset of the ring R. Further, we introduce the notion of generalized left bi-derivation and prove that if a prime ring R admits a generalized left bi-derivation G with associated left bi-derivation B then either R is commutative or G is a right bi-centralizer(or bi-multiplier) on R. Finally, it is shown that every generalized Jordan left derivation on a semisimple Banach algebra is continuous. Mathematics Subject Classification (2000). 16W25, 16N60, 46K15, 47B47. Keywords. Prime (semiprime) ring, semisimple Banach algebra, generalized left (right) derivation, generalized Jordan left (right) derivation, left (right) bi-derivation, generalized left (right) bi-derivation and left (right) bi-centralizer (bi-multiplier).

1. Introduction Throughout the present paper R will denote an associative ring with centre Z(R). A ring R is n-torsion free, where n > 1 is an integer, if nx = 0, x ∈ R implies x = 0. Recall that R is prime if aRb = (0) implies that a = 0 or b = 0. As usual [x, y] will denote the commutator xy − yx. We shall make use of commutator identities; [x, yz] = [x, y]z + y[x, z] and [xy, z] = [x, z]y + x[y, z]. Let S be a nonempty subset of R. A function f : R −→ R is said to be a centralizing function on S if [f (x), x] ∈ Z(R) for all x ∈ S. In the special case if [f (x), x] = 0 for all x ∈ S, f is said to be commuting on S. An additive mapping d : R −→ R is called a derivation (resp. Jordan derivation) if d(xy) = d(x)y + xd(y), (resp. d(x2 ) = d(x)x + xd(x)) holds for all x, y ∈ R. An additive mapping F : R −→ R is called a generalized derivation if there

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exists a derivation d : R −→ R such that F (xy) = F (x)y + xd(y) holds for all x, y ∈ R. An additive mapping δ : R −→ R is said to be a left derivation (resp. Jordan left derivation) if δ(xy) = xδ(y) + yδ(x) (resp. δ(x2 ) = 2xδ(x)) holds for all x, y ∈ R. An additive mapping δ : R −→ R is said to be a right derivation (resp. Jordan right derivation) if δ(xy) = δ(x)y + δ(y)x (resp. δ(x2 ) = 2δ(x)x) holds for all x, y ∈ R. If δ is both left as well as right derivation, then δ is a derivation. Clearly, every left(resp. right)derivation on a ring R is a Jordan left(resp. right) derivation but the converse need not be true in general; [see for example ([38], Example 1.1)]. In [5], Ashraf and Rehman proved that a Jordan left derivation on a 2-torsion free prime ring is a left derivation. Further in [6], the author together with Ashraf and Rehman showed that if R is a 2-torsion free prime ring and δ : R −→ R is an additive mapping such that δ(u2 ) = 2uδ(u) for all u in a square closed Lie ideal U of R, then either U ⊆ Z(R) or δ(U ) = (0). During the last 20 years, there has been ongoing interest concerning the relationship between left derivation and Jordan left derivation on prime and semiprime rings (cf.; [2,4,15,18,21,32,35,38] and reference therein). Following [39], an additive mapping H: R −→ R is called a left(resp. right) centralizer of R if H(xy) = H(x)y (resp. H(xy) = xH(y)) holds for all x, y ∈ R. An additive mapping H: R −→ R is called a Jordan left(resp. right) centralizer of R if H(x2 ) = H(x)x (resp. H(x2 ) = xH(x)) holds for all x ∈ R. Obviously, every left (resp. right) centralizer is a Jordan left (resp. right) centralizer. The converse is in general not true. In [39], Zalar proved that every Jordan left (resp. right) centralizer on a 2-torsion-free semiprime ring is a left (resp. right) centralizer. Let S be a subring of a ring R. A biadditive map (i.e., additive in both arguments) B : S × S −→ R is called a left(resp. right) bi-centralizer(or bi-multiplier) if B(xy, z) = B(x, z)y (resp. B(xy, z) = xB(y, z)) holds for all x, y, z ∈ S. A biadditive mapping B : S × S −→ R is called a Jordan left (resp. right) bi-centralizer if B(x2 , z) = B(x, z)x (resp. B(x2 , z) = xB(x, z)) holds for all x, z ∈ S. The map B is said to be a bi-centralizer if it is both a left and a right bi-centralizer on S. This type of map obviously extends the concepts of centralizers. In [26], Muthana presented an algebraic study of left bi-centralizers in prime rings. According to [4], an additive mapping G : R −→ R is called a generalized left derivation (resp. generalized Jordan left derivation)if there exists a Jordan left derivation δ : R −→ R such that G(xy) = xG(y) + yδ(x) (resp. G(x2 ) = xG(x) + xδ(x)) holds for all x, y ∈ R. It is easy to see that G: R −→ R is a generalized left derivation iff G is of the form G = δ + H, where δ is a left derivation and H a right centralizer on R. The concept of generalized left derivations cover the concept of left derivations. Moreover, a generalized left derivation with δ = 0 includes the concept of right centralizer(multiplier). Basic examples are left derivations and derivations(in the case of commutative rings). Since the sum of two generalized left derivations is a generalized

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left derivation, every map of the form G(x) = xa + δ(x), where a is a fixed element of R and δ is a left derivation of R, is a generalized left derivation. Notice that for any generalized left derivation G, the mapping F : R −→ R such that F (x) = G(x) + xa or F (x) = G(x) − xa, where a is a fixed element of R, is also a generalized left derivation on R. It is obvious to see that every generalized left derivation on a 2 -torsion free ring R is a generalized Jordan left derivation. But the converse need not be true in general (see Example 1.1 of [4]). It is shown in [4] that if R is a 2-torsion free prime ring, then every generalized Jordan left derivation on R is a generalized left derivation. Suppose that θ and φ are endomorphisms of R. An additive mapping d : R −→ R is called a (θ, φ)-derivation (resp. Jordan (θ, φ)-derivation) if d(xy) = d(x)θ(y) + φ(x)d(y) (resp. d(x2 ) = d(x)θ(x) + φ(x)d(x)) holds for all x, y ∈ R. Following [38], an additive mapping δ : R −→ R is called a left (θ, φ)derivation (resp. Jordan left (θ, φ)-derivation) if δ(xy) = θ(x)δ(y) + φ(y)δ(x) (resp. δ(x2 ) = θ(x)δ(x)+φ(x)δ(x)) holds for all x, y ∈ R. An additive mapping G : R −→ R is called a generalized left (θ, φ) -derivation (resp. generalized Jordan left (θ, φ)-derivation) if there exists a Jordan left (θ, φ) -derivation such that G(xy) = θ(x)G(y) + φ(y)δ(x) (resp. G(x2 ) = θ(x)G(x) + φ(x)δ(x)) holds for all x, y ∈ R. Note that for IR , the identity map on R, every generalized left (IR , IR )- derivation (resp. generalized Jordan left (IR , IR )-derivation) is a generalized left derivation (resp. generalized Jordan left derivation) on R. Clearly, this notion includes those left (θ, φ)-derivations when G = δ, left derivations when G = δ and θ = φ = IR , and generalized left derivations, which is the case when θ = φ = IR . In Sect. 3, besides proving some results concerning generalized Jordan left derivations in semiprime rings, we extend the above mentioned result ([4], Theorem 2.1) in the setting of semiprime rings. Further, some related results are also discussed. Let S be a non-empty subset of R and d : R −→ R be a derivation of R. If d(xy) = d(x)d(y) (resp. d(xy) = d(y)d(x)) holds for all x, y ∈ S, then d is said to act as a homomorphism (resp. anti-homomorphism) on S. In [8], Bell and Kappe proved that if K is a non-zero right ideal of a prime ring R and d : R −→ R is a derivation of R such that d acts as a homomorphism or an anti-homomorphism on K, then d = 0 on R. This result was further extended for left (θ, φ)-derivation in [2]. In Sect. 4, we study the generalized left derivation of a prime ring R which acts either as a homomorphism or as an anti-homomorphism on a certain well behaved subset of R. Let S be a subring of a ring R. A mapping B : R × R −→ R is said to be symmetric if B(x, y) = B(y, x) for all x, y ∈ R. Following [10], a biadditive map B : S × S −→ R is called a bi-derivation on S if it is a derivation in each argument, i.e., for every x ∈ S, the maps y −→ B(x, y) and y −→ B(y, x) are derivations of S into R (viz. [31], where bi-derivations satisfying some special properties are studied). Typical examples are mappings of the form (x, y) −→ c[x, y] where c is an element of the center of R. The

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notion of bi-derivation arises naturally in the study of additive commuting maps, since every commuting additive map f : S −→ R gives rise to a biderivation of S. Namely, linearization of the [f (x), x] = 0 for all x ∈ S yields that [f (x), y] = [x, f (y)] for all x, y ∈ S. Therefore, we note that the map (x, y) −→ [f (x), y] is a bi-derivation. The concept of bi-derivation was introduced by Maska [24]. Further, Bresar [10] showed that every bi-derivation B of a noncommutative prime ring R is of the form B(x, y) = λ[x, y] for some λ ∈ C, the extended centroid of R. Some results related to bi-derivations on prime and semiprime rings can be found in [1,3,16,31]. It is our attempt in Sect. 5 to initiate the study of left bi-derivations and generalized left bi-derivations, which are defined as follows: a biadditive mapping B : R × R −→ R is called a left bi-derivation(resp. Jordan left biderivation) if B(xy, z) = xB(y, z) + yB(x, z)(resp. B(x2 , z) = 2xB(x, z)) and B(z, xy) = xB(z, y)+yB(z, x)(resp. B(z, x2 ) = 2xB(z, x)) hold for all x, y, z ∈ R. A biadditive mapping G : R × R −→ R is called a generalized left bi-derivation (resp. generalized Jordan left bi-derivation) if there exists a Jordan left bi-derivation B : R × R −→ R such that G(xy, z) = xG(y, z) + yB(x, z)(resp. G(x2 , z) = xG(x, z) + xB(x, z)) and G(z, xy) = xG(z, y) + yB(z, x)(resp. G(z, x2 ) = xG(z, x) + xB(z, x)) holds for all x, y, z ∈ R. Hence, the concept of generalized left bi-derivations covers both the concepts of left bi-derivations and right bi-centralizers(bi-multipliers). Clearly, every right bi-centralizer(bimultiplier) is a generalized left bi-derivation on R. Thus, it is natural to ask whether every generalized left bi-derivation on a ring R is a right bicentralizer(bi-multiplier). It is shown in Sect. 5 that the answer to this question is affirmative under certain algebraic conditions. The last section of the present paper deals with the study of generalized Jordan left derivations on semisimple Banach algebras. The results we obtain may be of some interest from the automatic continuity point of view.

2. Preliminary results In this section we collect some known results and review a few important facts about the left Martindale ring of quotients that will be needed in the subsequent discussions. A ring Ql (R) will denote the left Martindale ring of quotients of a prime ring R. This ring was introduced by Martindale in [23] as a tool in the study of prime rings satisfying generalized polynomial identities (cf. [7]). It can be characterized by the following four properties (see [27], Proposition 10.2 for details): (i) R ⊆ Ql (R), (ii) for every q ∈ Ql (R) there exists a nonzero ideal I of R such that Iq ⊆ R, (iii) if q ∈ Ql (R) and I is a nonzero ideal of R such that Iq = 0, then q = 0,

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if I is an ideal of R and f : I −→ R is a left R-module map, then there exists q ∈ Ql (R) such that f (x) = xq for all x ∈ I. The center of Ql (R), will be denoted by C, and called the extended centroid of R. It is well known that C is a field. Also, it is easily seen that C is the centralizer of R in Ql (R). In particular, Z(R) ⊆ C. The subring of Ql (R) generated by R and C is called the central closure of R and will be denoted by RC . Another subring of Ql (R) is Qs (R) = {q ∈ Ql (R) : qI ⊆ R for some nonzero ideal I of R}. This ring is known as the symmetric Martindale ring of quotients. It is easy to verify that R ⊆ RC ⊆ Qs (R) ⊆ Ql (R). Note that aRb = (0) with a, b ∈ Ql (R) implies that a = 0 or b = 0. Whence we see that all RC , Ql (R) and Qs (R) are prime rings. Note that we have defined the notion of left derivation and generalized left derivation for the map f : R −→ R where R is any ring. However, this definition can be extended to the maps f : R −→ S in a natural way, where S is any ring, namely R, RC , R + RC , Ql (R), Qs (R), Ql (RC ), and Qs (RC )(see [7,27] for more details). We begin with the following lemmas which are essential for developing the proof of our main results.

(iv)

Lemma 2.1. [4, Lemma 2.2]. Let R be a 2-torsion free ring and G : R −→ R be a generalized Jordan left derivation with associated Jordan left derivation δ : R −→ R. Then (i) G(xy + yx) = xG(y) + yG(x) + xδ(y) + yδ(x) for all x, y ∈ R, (ii) G(xyx) = xyG(x) + 2xyδ(x) + x2 δ(y) − yxδ(x) for all x, y ∈ R, (iii) G(xyz + zyx) = xyG(z) + zyG(x) + 2xyδ(z) + 2zyδ(x)+ +xzδ(y) + zxδ(y) − yxδ(z) − yzδ(x) for all x, y, z ∈ R. Lemma 2.2. [5, Theorem]. Let R be a 2-torsion free prime ring and let U be a Lie ideal of R such that u2 ∈ U for all u ∈ U . If δ : R −→ R is an additive mapping such that δ(u2 ) = 2uδ(u) for all u ∈ U , then δ(uv) = uδ(v) + vδ(u) for all u, v ∈ U . Lemma 2.3. [13, Corollary 4.3]. Let A be a semisimple Banach algebra, let φ be an automorphism of A, and let d be a φ-derivation on A. Then d is continuous. Lemma 2.4. [17, Corollary 5]. Let R be a 2-torsion free semiprime ring. Then every Jordan derivation on R is a derivation. Lemma 2.5. [34, Lemma 1]. Let R be a semiprime ring. Suppose that the relation axb + bxc = 0 holds for all x ∈ R and some a, b, c ∈ R. In this case (a + c)xb = 0 for all x ∈ R. Lemma 2.6. [35, Theorem 2]. Let R be a 2-torsion free semiprime ring and δ : R −→ R be a Jordan left derivation. In this case δ is a derivation which maps R into Z(R).

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Lemma 2.7. [36, Theorem 4]. Let R be a 2-torsion free semiprime ring. Suppose that an additive mapping F : R −→ R satisfies the relation [[F (x), x], x] = 0 for all x ∈ R. In this case F is commuting. Lemma 2.8. [39, Proposition 1.4]. Let R be a semiprime ring with characteristic different from two and T : R −→ R be an additive mapping such that T (x2 ) = T (x)x(resp. T (x2 ) = xT (x)) for all x ∈ R. Then, T is a left (right) centralizer(multiplier). Lemma 2.9. [39, Corollary 1.5]. Let A be a semisimple Banach algebra and T : A −→ A be an additive mapping such that T (x2 ) = T (x)x(resp. T (x2 ) = xT (x)) for all x ∈ R. Then, T is a continuous linear operator. The following results are motivated by the work of Hvala ([19], Lemma 2). Proposition 2.10. Let R be a prime ring and f : R −→ RC be an additive map satisfying f (xy) = xf (y) for all x, y ∈ R. Then there exists q ∈ Ql (RC ) such that f (x) = xq for all x ∈ R. Proof. f is additive and satisfies f (xy) = xf (y) for all x, y ∈ R. This means that f is a left R-module homomorphism. Next,  the map f can be uniquely λi f (xi ), where extended from R to RC , say f¯. Indeed, we have f¯( λi xi ) = ¯ is well defined. It the map f xi ∈ R and λi ∈ C. Nowwe want to show that  λi f (xi ) = 0 for all xi ∈ R and is enough to check that λi xi = 0 implies λi ∈ C. Let I be a nonzero ideal in R such that Iλi ⊆ R for each i. Then, for every xi ∈ R we have     0=f aλi xi = (aλi )f (xi ) f ((aλi )xi ) =   =a λi f (xi ) for all a ∈ I.

(2.1)

 This implies that I( λi f (xi )) = (0). SinceR is prime and I is a nonzero ideal of R, the above expression yields that λi f (xi ) = 0. This proves f¯ to be a well defined homomorphism of left Ql (RC )-module. Hence, there exists q ∈ Ql (RC ) such that f¯(x) = xq for all x ∈ I. In particular, we find that f (x) = xq for all x ∈ R.  Proposition 2.11. Let R be a prime ring and h : R × R −→ RC be a biadditive map (i.e., additive in both arguments) satisfying h(xy, z) = xh(y, z) for all x, y, z ∈ R. Then there exists a function q from R into Ql (RC ) such that h(x, z) = xq(z) for all x, z ∈ R. Proof. The proof goes in the same way as the proof of Proposition 2.10 with the only exception that we take a biadditive map h instead of an additive map

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f. Also, instead of (2.1), we have for each xi , z ∈ R     0=h (aλi )xi , z = h((aλi )xi , z) = (aλi )h(xi , z)   =a λi h(xi , z) for all a ∈ I. and

     h(z, (aλi )xi ) = (aλi )h(z, xi ) 0 = h z, (aλi )xi =   =a λi h(z, xi ) for all a ∈ I.

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(2.2)

(2.3)

¯ ¯ z) for all x, y, z ∈ RC . This shows that This implies that h(xy, z) = xh(y, ¯ ¯ h : RC × RC −→ RC is a left Ql (RC )-module biadditive map and therefore h is well defined. Hence, there exists a function q from R into Ql (RC ) such that h(x, z) = xq(z) for all x, z ∈ R. 

3. Generalized Jordan left derivations on semiprime rings We shall start our investigations with our first result: Theorem 3.1. Let R be a 2-torsion free semiprime ring. If R admits a generalized Jordan left derivation G : R −→ R with associated Jordan left derivation δ : R −→ R, then δ is a derivation and [δ(x), y] = 0 for all x, y ∈ R. Proof. If R is commutative then there is nothing to prove. Therefore, we consider the noncommutative case only. Since G is a generalized Jordan left derivation with associated Jordan left derivation δ, by Lemma 2.1(i), we have G(xy + yx) = xG(y) + yG(x) + xδ(y) + yδ(x) for all x, y ∈ R.

(3.1)

Replacing y by xyx in (3.1), we get G(x2 yx + xyx2 ) = xG(xyx) + xyxG(x) + xδ(xyx) +xyxδ(x) for all x, y ∈ R. Using Lemma 2.1(ii) and the fact that δ(xyx) = x2 δ(y) + 3xyδ(x) − yxδ(x), we obtain G(x2 yx + xyx2 ) = x2 yG(x) + 5x2 yδ(x) + xyxG(x) +2x3 δ(y) − xyxδ(x) for all x, y ∈ R.

(3.2)

Again replacing y by xy + yx in Lemma 2.1(ii), we find that G(x(xy + yx)x) = G(x2 yx + xyx2 ) = x2 yG(x) + 4x2 yδ(x) + xyxG(x) + 2x3 δ(y) +xyxδ(x) − yx2 δ(x) for all x, y ∈ R.

(3.3)

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On combining (3.2) and (3.3), we get x2 yδ(x) − 2xyxδ(x) + yx2 δ(x) = 0 for all x, y ∈ R.

(3.4)

Here we want to mention that the proof runs on a similar approach with necessary variations as in equations (6) to (10) of [35]. But for the sake of completeness we present its details here. Replacing y by δ(x)y in (3.4), we obtain x2 δ(x)yδ(x) − 2xδ(x)yxδ(x) + δ(x)yx2 δ(x) = 0 for all x, y ∈ R.

(3.5)

Left multiplication by δ(x) of (3.4) yields that δ(x)x2 yδ(x) − 2δ(x)xyxδ(x) + δ(x)yx2 δ(x) = 0 for all x, y ∈ R.

(3.6)

From (3.5) and (3.6), we obtain [δ(x), x2 ]yδ(x) − 2[δ(x), x]yxδ(x) = 0 for all x, y ∈ R.

(3.7)

Replacing y by yx in (3.7), we obtain [δ(x), x2 ]yxδ(x) − 2[δ(x), x]yx2 δ(x) = 0 for all x, y ∈ R.

(3.8)

Multiplying (3.7) by x from the right, we get [δ(x), x2 ]yδ(x)x − 2[δ(x), x]yxδ(x)x = 0 for all x, y ∈ R.

(3.9)

Combining (3.8) and (3.9), we find that [δ(x), x2 ]y[δ(x), x] + [δ(x), x]y(−2x[δ(x), x]) = 0 for all x, y ∈ R. (3.10) Application of Lemma 2.5 yields that ([δ(x), x2 ] − 2x[δ(x), x])y[δ(x), x] = 0 for all x, y ∈ R and hence [[δ(x), x], x]y[δ(x), x] = 0 for all x, y ∈ R. This implies that [[δ(x), x], x]y[[δ(x), x], x] = 0 for all x, y ∈ R. The semiprimeness of R gives that [[δ(x), x], x] = 0 for all x ∈ R. In view of Lemma 2.7, we obtain [δ(x), x] = 0 for all x ∈ R.

(3.11)

That is, δ is commuting on R. Since δ is a Jordan left derivation and commuting on R, δ is a Jordan derivation. Thus, δ is a derivation by Lemma 2.4. It is well known and easy to prove that any commuting derivation on a semiprime ring R maps R into Z(R). Therefore, we have [δ(x), y] = 0 for all x, y ∈ R. This completes the proof of our theorem.  In the following there are some immediate consequences of the above theorem. Corollary 3.2. [35, Theorem 2]. Let R be a 2-torsion free semiprime ring, and let δ : R −→ R be a Jordan left derivation. In this case δ is a derivation which maps R into Z(R).

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Corollary 3.3. Let R be a 2-torsion free prime ring. If R admits a generalized Jordan left derivation with an associated nonzero Jordan left derivation δ, then R is commutative. Proof. Let G : R −→ R be a generalized Jordan left derivation with an associated Jordan left derivation δ : R −→ R. Then, Theorem 3.1 asserts that δ is a derivation and [δ(x), y] = 0 for all x, y ∈ R. This implies that [δ(x), x] ∈ Z(R) for all x ∈ R. Since R is prime and δ is a nonzero derivation, R is commutative by Theorem 2 of [28].  Corollary 3.4. Let R be a noncommutative 2-torsion free prime ring. If R admits a generalized Jordan left derivation G : R −→ R with associated Jordan left derivation δ : R −→ R, then G(x) = qx for all x ∈ R and some q ∈ Ql (RC ). Proof. Since R is a noncommutative 2-torsion free prime ring and G is a generalized Jordan left derivation with associated Jordan left derivation δ on R, in view of Corollary 3.3, we have δ = 0. Thus, we obtain G(x2 ) = xG(x) for all x ∈ R. That is, G is a Jordan right centralizer and therefore in view of Lemma 2.8, G is a right centralizer on R i.e., G(xy) = xG(y) for all x, y ∈ R. Hence, there exists q ∈ Ql (RC ) such that G(x) = xq for all x ∈ R by Proposition 2.10.  Corollary 3.5. Let R be a 2-torsion free prime ring, and let G : R −→ R be a generalized Jordan left derivation with an associated nonzero Jordan left derivation δ : R −→ R. In this case G is a generalized derivation on R. Proof. In view of Theorem 3.1 and Corollary 3.3, δ is a derivation and R is commutative. Since R is a 2-torsion free prime ring and G is a generalized Jordan left derivation, by Theorem 3.2 of [4], G is a generalized left derivation. Thus, we have G(yx) = yG(x) + xδ(y) = G(x)y + xδ(y) for all x, y ∈ R. This implies that G(xy) = G(x)y + xδ(y) for all x, y ∈ R. Hence G is, indeed, a generalized derivation on R. This proves the corollary.  The next result is an extension of Theorem 3.2. in [4]. Theorem 3.6. Let R be a 2-torsion free semiprime ring, and let G : R −→ R be a generalized Jordan left derivation with associated Jordan left derivation δ : R −→ R. Then every generalized Jordan left derivation is a generalized left derivation on R. Proof. We break up the proof into two steps. Step 1. If the associated Jordan left derivation δ = 0, then G is a Jordan right centralizer on R. Therefore, in view of Lemma 2.8, G is a right centralizer. Hence for δ = 0, G is a generalized left derivation.

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Step 2. On the other hand, suppose that the associated Jordan left derivation δ= 0. Then in view of Lemma 2.6, δ is a derivation (i.e., δ is a left as well as a right derivation on R). Since G, H and δ are additive maps on R, we write H = G − δ. Then, we find that H(x2 ) = (G − δ)(x2 ) = G(x2 ) − δ(x2 ) = xG(x) + xδ(x) − xδ(x) − xδ(x) = xG(x) − xδ(x) = x(G(x) − δ(x)) = xH(x) for all x ∈ R. This implies that H(x2 ) = xH(x) for all x ∈ R. That is, H is a Jordan right centralizer on R. Thus, by Lemma 2.8, one can conclude that H is a right centralizer. Therefore, we prove that G can be written as G = H + δ, where δ is a left derivation and H is a right centralizer on R. Hence, G is a generalized left derivation on R. This proves the theorem completely.  We can derive some interesting corollaries from Theorem 3.6. Corollary 3.7. Let R be a 2-torsion free semiprime ring and δ : R −→ R be a Jordan left derivation. Then every Jordan left derivation is a left derivation on R. Corollary 3.8. Let R be a 2-torsion free semisimple ring, and let G : R −→ R be a generalized Jordan left derivation with associated Jordan left derivation δ : R −→ R. Then every generalized Jordan left derivation is a generalized left derivation on R. Proof. It is the consequence of Theorem 3.6 and of the fact that every semisimple ring is semiprime.  The following example demonstrates that it is essential for R to be semiprime in the hypotheses of Theorem 3.6. Example 3.9. Let S be a ring such that the square of each element in S is zero,⎧but product of⎫some nonzero elements in S is nonzero. Next, ⎛ the ⎞ ⎨ 000 ⎬ let R= ⎝ a 0 0 ⎠ |a, b ∈ S . Define the maps G, δ : R −→ R such that ⎩ ⎭ b⎞ a0 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ 0 0 0 0 0 0 0 0 0 0 0 0 G ⎝ a 0 0 ⎠ = ⎝ 0 0 0 ⎠ and δ ⎝ a 0 0 ⎠ = ⎝ a 0 0 ⎠ . It is b a 0 b 0 0 b a 0 0 a 0 straightforward to check that G is a generalized Jordan left derivation. However, G is not a generalized left derivation.

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4. Generalized left derivations which act as homomorphism or as anti-homomorphism Let S be a nonempty subset of R and G be a generalized left derivation on R. If G(xy) = G(x)G(y) or G(xy) = G(y)G(x) for all x, y ∈ S, then G is said to be a generalized left derivation which acts as a homomorphism or an anti-homomorphism on S, respectively. Of course, derivations which act as endomorphisms or anti-endomorphisms of a ring R may behave as such on certain subsets of R, for example, any derivation d behaves as the zero endomorphism on the subring C consisting of all constants (i.e., elements x for which d(x) = 0). In fact, in a semiprime ring R, d may behave as an endomorphism on a proper ideal of R. As an example of such R and d, let S be any semiprime ring with a nonzero derivation δ, take R = S ⊕ S and define d by d(r1 , r2 ) = (δ(r1 ), 0). However, in the case of prime rings, Bell and Kappe in [8] showed that the behaviour of d is somewhat more restricted. By proving that if R is a prime ring and d is a derivation of R which acts as a homomorphism or an anti-homomorphism on a nonzero right ideal of R, then d = 0 on R. Further, Ashraf obtained in [2] the above mentioned result for left (θ, φ)-derivations as follows: Let R be a prime ring and I be a nonzero ideal of R. Suppose θ and φ are endomorphisms of R and δ : R −→ R is a left (θ, φ)-derivation of R. If δ acts as a homomorphism or as an anti-homomorphism on I, then δ = 0 on R. In the present section, our objective is to extend the above study to the setting of generalized left derivations. In fact, we prove the following result: Theorem 4.1. Let R be a 2-torsion free prime ring, and let I be a nonzero ideal of R. Suppose that G : R −→ R is a generalized left derivation with associated Jordan left derivation δ : R −→ R. (i) (ii)

If G acts as a homomorphism on I, then either R is commutative or G(x) = xq for all x ∈ R and q ∈ Ql (RC ). If G acts as an anti-homomorphism on I, then either R is commutative or G(x) = xq for all x ∈ R and q ∈ Ql (RC ).

Proof. (i) First we suppose G acts as a homomorphism, then by the hypothesis we have G(xy) = G(x)G(y), which can be rewritten as G(xy) = G(x)G(y) = xG(y) + yδ(x) for all x, y ∈ I.

(4.1)

Then, on the one hand we have G(xyz) = G(x(yz)) = xG(yz) + yzδ(x) = xG(y)G(z) + yzδ(x) for all x, y, z ∈ I.

(4.2)

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On the other hand, we find that G(xyz) = G((xy)z) = G(xy)G(z) = xG(y)G(z) + yδ(x)G(z) for all x, y, z ∈ I.

(4.3)

Combining (4.2) and (4.3), we obtain yzδ(x) = yδ(x)G(z) for all x, y, z ∈ I. This implies that y(zδ(x) − δ(x)G(z)) = 0 for all x, y, z ∈ I, that is, (zδ(x) − δ(x)G(z))IR(zδ(x) − δ(x)G(z))I = (0) for all x, z ∈ I. Thus, the primeness of R forces that (zδ(x) − δ(x)G(z))I = (0) for all x, z ∈ I. The last expression implies that zδ(x) − δ(x)G(z) = 0 for all x, z ∈ I.

(4.4)

Since R is a 2-torsion free prime ring and I is a nonzero ideal of R, δ is a left derivation by Lemma 2.2. Replacing x by xy in the last expression and using the fact that δ is a left derivation, we find that [z, x]δ(y) + [z, y]δ(x) = 0 for all x, y, z ∈ I. Taking x = z, we obtain [x, y]δ(x) = 0 for all x, y ∈ I. Further, replace y by ry in the last relation to get [x, r]yδ(x) = 0 for all x, y ∈ I and r ∈ R. That is, [x, r]Rzδ(x) = (0) for all x, z ∈ I and r ∈ R. The primeness of R forces that either [x, r] = 0 or zδ(x) = 0. Now, we put I1 = {x ∈ I | [x, r] = 0 for all r ∈ R} and I2 = {x ∈ I | zδ(x) = 0 for all z ∈ I}. Then, clearly each of I1 and I2 are additive subgroups of I. Moreover, by the discussion given, I is the set-theoretic union of I1 and I2 . But a group cannot be the set-theoretic union of its two proper subgroups, hence I1 = I or I2 = I. If I1 = I, then [x, r] = 0 for all x ∈ I and r ∈ R. This implies that I is a central ideal and hence R is commutative. On the other hand, if I2 = I, then zδ(x) = 0 for all x, z ∈ I. That is, IRδ(x) = (0) for all x ∈ I. Since R is prime and I is a nonzero ideal of R, the last relation yields that δ(x) = 0 for all x ∈ I. Replacing x by xr, we obtain xδ(r) = 0 for all x ∈ I and r ∈ R, i.e., IRδ(r) = (0) for all r ∈ R. Thus, the primeness of R forces that δ = 0 on R. Hence, there exists q ∈ Ql (RC ) such that G(x) = xq for all x ∈ R by Proposition 2.10. (ii) Finally we assume that G acts as an anti-homomorphism, then we have G(xy) = G(y)G(x) for all x, y ∈ I. This can be rewritten as xG(y) + yδ(x) = G(y)G(x) for all x, y ∈ I.

(4.5)

Replacing y by xy in (4.5), we get xG(xy) + xyδ(x) = xG(y)G(x) + yδ(x)G(x) for all x, y ∈ I.

(4.6)

In view of (4.5), the last expression yields that xyδ(x) = yδ(x)G(x) for all x, y ∈ I.

(4.7)

Again, replace y by ry in (4.7) to get xryδ(x) = ryδ(x)G(x) for all x, y ∈ I and r ∈ R.

(4.8)

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Left multiplication by r of (4.7), gives that rxyδ(x) = ryδ(x)G(x) for all x, y ∈ I and r ∈ R.

(4.9)

Combining (4.8) and (4.9), we find that [x, r]yδ(x) = 0 for all x, y ∈ I and r ∈ R. Notice that the arguments given in the last paragraph of the proof of Part (i) above are still valid in the present situation and hence repeating the same process, we get the required result. The theorem is now proved.  Corollary 4.2. Let R be a 2-torsion free prime ring and I be a nonzero ideal of R. Suppose that δ is a Jordan left derivation. (i) If δ acts as a homomorphism on I, then either R is commutative or δ = 0 on R. (ii) If δ acts as an anti-homomorphism on I, then either R is commutative or δ = 0 on R.

5. Generalized left bi-derivations on prime rings Over the last few decades, several authors have investigated the relationship between the commutativity of the ring R and certain specific types of derivations of R. The first result in this direction is due to Posner [28] who proved that if a prime ring R admits a nonzero derivation d such that [d(x), x] ∈ Z(R) for all x ∈ R, then R is commutative. This result was subsequently refined and extented by a number of algebraists; we refer to ([9,16] where further references can be found) for a state-of-the art account and a comprehensive bibliography. In [15], Bresar and Vukman showed that a prime ring must be commutative if it admits a nonzero left derivation. In the present section, our aim is to extend the above mentioned result in the setting of generalized left bi-derivation. The result states as follows: Theorem 5.1. Let R be a prime ring. If R admits a generalized left bi-derivation G : R × R → R with associated left bi-derivation B : R × R → R, then either R is commutative or G is a right bi-centralizer(bi-multiplier) on R. Proof. We are given that G is a generalized left bi-derivation, we have G(xy, z) = xG(y, z) + yB(x, z) for all x, y, z ∈ R.

(5.1)

Replacing y by yw in (5.1), we obtain G(x(yw), z) = xyG(w, z) + xwG(y, z) + ywB(x, z) for all w, x, y, z ∈ R. (5.2)

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Also, we have G((xy)w, z) = xyG(w, z) + wxB(y, z) + wyB(x, z) for all x, y, z ∈ R. (5.3) Combining (5.2) and (5.3), we find that [x, w]B(y, z) + [y, w]B(x, z) = 0 for all w, x, y, z ∈ R.

(5.4)

Substituting w for y in the last relation, we find that [x, w]B(w, z) = 0 for all w, x, z ∈ R

(5.5)

Linearization to z yields that [x, w]B(w, z) + [x, w]B(w, t) = 0 for all w, x, z, t ∈ R.

(5.6)

The application of relation (5.5) gives that [x, w]B(w, t) = 0 for all w, x, t ∈ R. This implies that [z, w]xB(w, t) = 0 for all w, x, z, t ∈ R, i.e., [z, w]RB(w, t) = (0) for all w, z, t ∈ R. Thus, for each w ∈ R, either [z, w] = 0 or B(w, t) = 0. Now let U1 = {w ∈ R | [z, w] = 0 for all z ∈ R} and U2 = {w ∈ R | B(w,  t) = 0 for all t ∈ R}. Then U1 , U2 are both additive subgroups of R and U1 U2 = R. By Brauer’s trick, we have either R = U1 or R = U2 . On the one hand, if R = U1 , then [z, w] = 0 for all w, z ∈ R. Therefore, R is commutative. On the other hand, if R = U2 , then we have B(w, t) = 0 for all w, t ∈ R and hence we get the required result. This completes the proof of the theorem.  Corollary 5.2. Let R be a prime ring. If R admits a generalized left bi-derivation G : R × R → R with associated nonzero left bi-derivation B : R × R → R, then R is commutative. Corollary 5.3. Let R be a prime ring. If R admits a left bi-derivation B : R × R → R, then either R is commutative or B = 0. Corollary 5.4. Let R be a noncommutative prime ring. If R admits a generalized left bi-derivation G : R × R → R with associated left bi-derivation B : R × R → R, then G(x, z) = xq(z) for all x, z ∈ R and some function q from R into Ql (R). Proof. Since R is a noncommutative prime ring and G is a generalized left bi-derivation of R, we have G(xy, z) = xG(y, z) for all x, y, z ∈ R by Theorem 5.1. Hence, in view of Proposition 2.11 we get the required result.  It would be interesting to know whether Corollary 5.2 holds in the case of arbitrary rings. The following example justifies the fact. ⎧⎛ ⎫ ⎞ ⎨ 0ab ⎬ Example 5.5. Let S be any ring. Next, consider R= ⎝ 0 0 c ⎠ | a, b, c ∈ S . ⎩ ⎭ 000 Define the maps G, B : R×R −→ R such that G(x, y) = B(x, y) = [x, y] for all

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x, y ∈ R. Then, G and B satisfy the requirements of Corollary 5.2. However, R is not commutative.

6. Generalized Jordan left derivations on Banach algebras We begin this section with the following theorem. Theorem 6.1. Let A be a semisimple Banach algebra, and let G : A −→ A be a generalized Jordan left derivation with associated Jordan left derivation δ : A −→ A. Then every generalized Jordan left derivation on A is continuous. Let us explain in somewhat more details the background and motivation of Theorem 6.1. In the year 1955, Singer and Wermer [29] proved that a continuous linear derivation on a commutative Banach algebra maps the algebra into its radical. Johnson and Sinclair [20] proved that any linear derivation on a semisimple Banach algebra is continuous. According to these two results, one can conclude that there are no nonzero linear derivations on a commutative semisimple Banach algebra. Singer and Wermer conjectured in [29] that the continuity assumption in their result is superfluous. It took more than 30 years until this conjecture was finally proved by Thomas [30]. Obviously, from Thomas’s result it follows directly that there are no nonzero linear derivations on a commutative semisimple Banach algebra. By our knowledge the first noncommutative extension of the Singer–Wermer theorem was proved by Yood [37] by showing that if for all pairs x, y ∈ A, where A is a noncommutative Banach algebra, the element [d(x), y] ∈ rad(A), then d maps A into rad(A). Further, Bresar and Vukman [14] generalized Yood’s result by proving that in the case [d(x), x] ∈ rad(A) for all x ∈ A, d maps A into rad(A). More related results were also shown in [11–13,22,25,33] where further references can be found. Proof of Theorem 6.1. Notice that every semisimple Banach algebra is a semiprime ring. If the associated Jordan left derivation δ = 0, then G is a Jordan right multiplier on A. Therefore, in view of Lemma 2.9, G is continuous. Hence for δ = 0, every generalized Jordan left derivation is continuous (linear operator).  On the other hand, suppose that the associated Jordan left derivation δ= 0. In view of Lemma 2.6, the associated Jordan left derivation δ is a linear derivation on A. Thus by Lemma 2.3, δ is continuous. Let us set H = G − δ. Then H is a Jordan right centralizer on A and hence H is a right centralizer on A. That is, H(xy) = xH(y) for all x, y ∈ A. Hence in view of Lemma 2.9, we conclude that H is continuous. This implies that G is continuous, as desired. The proof of the theorem is complete.

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As consequences of Theorem 6.1, we obtain the following corollaries: Corollary 6.2. Let A be a semisimple Banach algebra. Then every Jordan left derivation on A is continuous. Corollary 6.3. Let A be a semisimple Banach algebra. Then every right centralizer on A is continuous. Remark 6.4. The concepts of generalized Jordan left derivation and generalized Jordan right derivation are symmetric, therefore it is obvious that every generalized Jordan right derivation on a semisimple Banach algebra is continuous.

Acknowledgments The author would like to thank Professors M. Bresar and M. Ashraf for useful discussion. Substantial part of this work was done when the author was a visiting scientist at Harish-Chandra Research Institute (HRI), Allahabad (December 2008). The final form was prepared when the author was on a short visit at the Department of Mathematics, Indian Institute of Science (IISc), Bangalore (June 2009). The author appreciates the gracious hospitality he received at each institution during his visits.

References [1] Argac, N., Yenigul, M.S.: Lie ideal and symmetric bi-derivations of prime rings. Pure Appl. Math. Sci. XLIV(1–2), 17–21 (1996) [2] Ashraf, M.: On left (θ, φ)-derivations of prime rings. Arch. Math. (Brno) 4, 157–166 (2005) [3] Ashraf, M.: On symmetric biderivations in rings. Rend. Instit. Mat. Trieste XXXI, 25–36 (1999) [4] Ashraf, M., Ali, S.: On generalized Jordan left derivations in rings. Bull. Korean. Math. Soc. 45(2), 253–261 (2008) [5] Ashraf, M., Rehman, N.: On Lie ideals and Jordan left derivations of prime rings. Arch. Math. (Brno) 36, 201–206 (2000) [6] Ashraf, M., Rehman, N., Ali, S.: On Jordan left derivations of Lie ideals in prime rings. Southeast Asian Bull. Math. 25, 379–382 (2001) [7] Beidar, K.I., Matindale III, W.S., Mikhalev, A.V.: Ring with generalized identities. In: Monographs and Textbooks in pure and applied mathematics. Marcel Dekker, New York (1995) [8] Bell, H.E., Kappe, L.C.: Ring in which derivations satisfying certain algebraic conditions. Acta Math. Hungar 53, 339–346 (1989) [9] Bell, H.E., Martindale III, W.S.: Centralizing mappings of semiprime rings. Canad. Math. Bull. 30, 92–101 (1987) [10] Bresar, M.: On generalized biderivations and related maps. J. Algebra 172, 764–786 (1995) [11] Bresar, M.: Derivations of noncommutative Banach algebras II. Arch. Math. 63, 56–59 (1994)

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[12] Bresar, M.: Jordan derivations on semiprime rings. Proc. Am. Math. Soc. 104(4), 1003–1006 (1988) [13] Bresar, M., Villena, A.R.: The noncommutative Singer–Wermer conjecture and φ-derivations. J. Lond. Math. Soc. 66, 710–720 (2002) [14] Bresar, M., Vukman, J.: Derivations of noncommutative Banach algebras. Arch. Math. 59, 367–370 (1992) [15] Bresar, M., Vukman, J.: On left derivations and related mappings. Proc. Am. Math. Soc. 110, 7–16 (1990) [16] Bresar, M., Martindale III, W.S., Miers, C.R.: Centralizing maps in prime rings with involution. J. Algebra 161, 342–357 (1993) [17] Cusack, J.M.: Jordan derivations on rings. Proc. Am. Math. Soc. 53, 321–324 (1975) [18] Deng, Q.: On Jordan left derivations. Math. J. Okayama Univ. 34, 145–147 (1992) [19] Hvala, B.: Generalized derivations in rings. Comm. Algebra 26, 1147–1166 (1998) [20] Johnson, B.E., Sinclair, A.M.: Continuity of derivations and a problem of Kaplansky. Am. J. Math. 90, 1068–1073 (1968) [21] Jun, K.-W., Kim, B.D.: A note on Jordan left derivations. Bull. Korean Math. Soc. 33, 221–228 (1996) [22] Kim, B.: On derivations on semiprime rings and noncommutative Banach algebras, Acta Math. Sinica, English Series 16 (2000) [23] Martindale III, W.S.: Prime rings satisfying a generalized polynomial identity. J. Algebra 12, 576–584 (1969) [24] Maska, Gy.: Remark on symmetric bi-additive functions having non-negative diagonalization. Glasnik Matematicki 15, 279–280 (1980) [25] Mathieu, M., Murphy, G.J.: Derivations mapping into radical. Arch. Math. 57, 469–474 (1991) [26] Muthana, N.M.: Left centralizer traces, generalized biderivations, left bi-multipliers and generalized Jordan biderivations. Aligarh Bull. Math. 26(2), 33–45 (2007) [27] Passman, D.: Infinite Crossed Products. Academic Press, San Diego (1989) [28] Posner, E.C.: Derivations in prime rings. Proc. Am. Math. Soc. 8, 1093–1100 (1957) [29] Singer, I.M., Wermer, J.: Derivations on commutative normed algebras. Math. Ann. 129, 260–264 (1955) [30] Thomas, M.P.: The image of derivations is contained in the radical. Ann. Math. 128, 435–460 (1988) [31] Vukman, J.: Two results concerning symmetric biderivations on prime rings. Aequationes Math. 40, 181–189 (1990) [32] Vukman, J.: Jordan left derivations on semiprime rings. Math. J. Okayama Univ. 39, 1–6 (1997) [33] Vukman, J.: On some additive mapping in semiprime rings and Banach algebras. Aequationes Math. 58, 1–10 (1999) [34] Vukman, J.: Centralizers on semiprime rings. Comment. Math. Univ. Carolinae 42, 237–245 (2001) [35] Vukman, J.: On left Jordan derivations of rings and Banach algebras. Aequationes Math. 75, 260–266 (2008) [36] Vukman, J., Ulbl, I.K.: On some equations related to derivations in rings. Int. J. Math. Math. Sci. 17, 2703–2710 (2005) [37] Yood, B.: Continuous homomorphisms and derivations on Banach algebras. Contemp. Math. 32, 279–284 (1984) [38] Zaidi, S.M.A., Ashraf, M., Ali, S.: On Jordan ideals and left (θ, θ)-derivations in prime rings. Int. J. Math. Math. Sci. 37, 1957–1964 (2004) [39] Zalar, B.: On centralizer of semiprime rings. Comm. Math. Univ. Carolinae 32, 609–614 (1991)

226 Shakir Ali Department of Mathematics Aligarh Muslim University Aligarh 202002 India e-mail: shakir50@rediffmail.com Received: July 31, 2009 Revised: September 19, 2010

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