International Journal of Algebra, Vol. 5, 2011, no. 1, 17 - 23 A Note on Generalized Derivations of Prime Rings Neşet AYDIN Çanakkale Onsekiz Mart University Faculty of Arts and Science Department of Mathematics 17020 Çanakkale, Turkey neseta@comu.edu.tr Abstract In the present paper we prove the following result; Let R be a noncommutative prime ring, I an ideal of R, (F, d) a generalized derivation of R and a ∈ R. If F ([x, a]) = 0 or [F (x), a] = 0 for all x ∈ I, then, d(x) = λ[x, a] for all x ∈ I or a ∈ Z. Mathematics Subject Classification: 16N60, 16W25, 16U80 Keywords: Derivations, Generalized Derivations, Prime Rings, Ideals 1 Introduction Throughout R will be a ring with center Z. Let x, y ∈ R. The commutator xy − yx will be denoted by [x, y]. Recall that a ring is prime if xRy = 0 implies x = 0 or y = 0. An additive mapping ∂ : R → R is called a derivation if ∂(xy) = ∂(x)y + x∂(y) holds for all x, y ∈ R. The study of commutativity of prime rings with derivation was initiated by Posner [6]. During the past few decades, there has been an ongoing interest concerning the relationship between the commutativity of a ring and the existence of certain specific types of derivations of R. Recently, in [2], Bresar defined the following notation. An additive mapping F : R → R is called a generalized derivation if there exists a derivation d : R → R such that F (xy) = F (x)y + xd(y), for all x, y ∈ R. Throughout in this paper, R will be a prime ring with Martindale ring of quotients Qr (R), extended centroid C and central closure RC = RC (see [5, sec. 10], for detail). In [3], B. Havala studied generalized derivation can be uniquely extended to a generalized derivation of Qr (R). In [1], Albas, E. and Argac, N. showed that R is a non-commutative ring and d a generalized derivation 18 Neşet AYDIN determined by a derivation α of R, a ∈ R and for all x ∈ R, [a, d(x)] = 0 then either a ∈ C or there exist λ, η ∈ C such that d(x) = ηx + λ(ax + xa), for all x ∈ R. In this paper, our aim is to extend this result to a non-zero ideal I of R. In particular, our research can be viewed as a new more elementary approach. 2 Main Results In the following, we assume that R is a prime ring. We denote a generalized derivation F : R → R determined by a derivation d of R by (F, d). Lemma 2.1 Let R be a non-commutative ring, I an ideal of R and (F, d) a generalized derivation of R and a ∈ R. If a ∈ / Z and [F (x), a] = 0 for all x ∈ I, then F ([x, a]) = 0 for all x ∈ I. Proof. We replace x by xr, r ∈ R in the defining equation [F (x), a] = 0 for all x ∈ I (1) to obtain, 0 = [F (xr), a] = [F (x)r + xd(r), a] = [F (x)r, a] + [xd(r), a] = F (x)[r, a] + [F (x), a]r + x[d(r), a] + [x, a]d(r) for all x ∈ I, r ∈ R which implies that F (x)[r, a] + x[d(r), a] + [x, a]d(r) = 0 (2) for all x ∈ I, r ∈ R. In (2), replace x by xy, y ∈ I and use (2) , we obtain 0 = F (xy)[r, a] + xy[d(r), a] + x[y, a]d(r) + [x, a]yd(r) = F (x)y[r, a] + xd(y)[r, a] + xy[d(r), a] + x[y, a]d(r) + [x, a]yd(r) = F (x)y[r, a] + xd(y)[r, a] + x(y[d(r), a] + [y, a]d(r)) + [x, a]yd(r) = F (x)y[r, a] + xd(y)[r, a] − xF (y)[r, a] + [x, a]yd(r) = (F (x)y + xd(y) − xF (y))[r, a] + [x, a]yd(r) so we get (F (x)y + xd(y) − xF (y))[r, a] + [x, a]yd(r) = 0, ∀x, y ∈ I, r ∈ R Replace r by a in (3), we have [x, a]Id(a) = 0, ∀x ∈ I (3) Generalized Derivations of Prime Rings 19 Since a ∈ / Z and the primeness of I, yields d(a) = 0 If we substitute sx, s ∈ R for x in (3), then we get 0 = (F (sx)y + sxd(y) − sxF (y))[r, a] + [sx, a]yd(r) = ((F (s)x + sd(x))y + sxd(y) − sxF (y))[r, a] + s[x, a]yd(r) + [s, a]xyd(r) = (F (s)xy + sd(x)y + sxd(y) − sxF (y))[r, a] + s[x, a]yd(r) + [s, a]xyd(r) = F (s)xy[r, a] + sd(x)y[r, a] + sxd(y)[r, a] − sxF (y)[r, a] + s[x, a]yd(r) + [s, a]xyd(r) = (F (s)xy + sd(x)y)[r, a] + s ((xd(y) − xF (y))[r, a] + [x, a]yd(r)) + [s, a]xyd(r) = (F (s)xy + sd(x)y)[r, a] + s(−F (x)y[r, a]) + [s, a]xyd(r) = (F (s)xy + sd(x)y − sF (x)y)[r, a] + [s, a]xyd(r) and so (F (s)x + sd(x) − sF (x))y[r, a] + [s, a]xyd(r) = 0 (4) In (4) replacing s by a, (F (a)x + ad(x) − aF (x))y[r, a] = 0 ∀x, y ∈ I (5) Using a ∈ / Z and the primeness of I, we obtain F (a)x + ad(x) − aF (x) = 0. Then we have F (ax) = aF (x), ∀x ∈ I (6) On the other hand, by employing d(a) = 0, we see that the relation F (xa) = F (x)a + xd(a) = F (x)a is reduced to F (xa) = F (x)a, ∀x ∈ I Combining (6) and (7), we arrive at F ([x, a]) = F (xa) − F (ax) = F (x)a − aF (x) By using the hypothesis, we have F ([x, a]) = [F (x), a] = 0, ∀x ∈ I This completes the proof. (7) 20 Neşet AYDIN Lemma 2.2 Let R be a non-commutative prime ring, I an ideal of R, (F, d) a generalized derivation of R and a ∈ R. If a ∈ / Z and F ([x, a]) = 0 for all x ∈ I, then [F (x), a] = 0 for all x ∈ I. Proof. we replace x by xa in the defining equation F ([x, a]) = 0 to obtain 0 = F ([xa, a]) = F ([x, a]a) = F ([x, a])a + [x, a]d(a) and so [x, a]d(a) = 0, for all x ∈ I (8) Taking xy, y ∈ I instead of x in (8), 0 = [xy, a]d(a) = x[y, a]d(a) + [x, a]yd(a) and using (8) we obtain [x, a]Id(a) = 0, for all x ∈ I (9) By the primeness of I and a ∈ / Z, (9) implies that d(a) = 0 Now we replace x by xy, y ∈ I in the defining equation F ([x, a]) = 0 to obtain 0 = F ([xy, a]) = F (x[y, a] + [x, a]y) = F ([x, a]y) + F (x[y, a]) = F ([x, a])y + [x, a]d(y) + F (x)[y, a] + xd([y, a]) = [x, a]d(y) + F (x)[y, a] + x([d(y), a] + [y, d(a)]) Since d(a) = 0, we have F (x)[y, a] + [x, a]d(y) + x[d(y), a] = 0, ∀x, y ∈ I (10) Substitute yz, z ∈ I instead of y in (10) and use (10), we arrive at 0 = F (x)[yz, a] + [x, a]d(yz) + x[d(yz), a] = F (x)y[z, a] + F (x)[y, a]z + [x, a]d(y)z + [x, a]yd(z) + x[d(y)z, a] + x[yd(z), a] = F (x)y[z, a] + (F (x)[y, a] + [x, a]d(y))z + [x, a]yd(z) + xd(y)[z, a] + x[d(y), a]z + xy[d(z), a] + x[y, a]d(z) = F (x)y[z, a] + (F (x)[y, a] + [x, a]d(y) + x[d(y), a])z + [x, a]yd(z) + xd(y)[z, a] + xy[d(z), a] + x[y, a]d(z) = F (x)y[z, a] + [x, a]yd(z) + xd(y)[z, a] + xy[d(z), a] + x[y, a]d(z) = (F (x)y + xd(y))[z, a] + [x, a]yd(z) + x(y[d(z), a] + [y, a]d(z)) = (F (x)y + xd(y))[z, a] + [x, a]yd(z) − xF (y)[z, a] Generalized Derivations of Prime Rings 21 and so (F (x)y + xd(y) − xF (y))[z, a] + [x, a]yd(z) = 0, ∀x, y, z ∈ I (11) Replace x by ax in (11) and use (11), it yields 0 = (F (ax)y + axd(y) − axF (y))[z, a] + a[x, a]yd(z) = F (ax)y[z, a] + a(xd(y)[z, a] − xF (y)[z, a] + [x, a]yd(z)) = F (ax)y[z, a] − aF (x)y[z, a] Hence we get (F (ax) − aF (x))y[z, a] = 0, ∀x, y, z ∈ I (12) Since a ∈ / Z and the primeness of I, we have F (ax) = aF (x), ∀x ∈ I (13) On the other hand, since d(a) = 0, F (xa) = F (x)a + xd(a) = F (x)a (14) Combining (13) and (14) we arrive at [F (x), a] = F (x)a − aF (x) = F (xa) − F (ax) = F ([x, a]) = 0 and so [F (x), a] = 0, ∀x ∈ I Thus. the proof is complete. The following theorem is motivated by [1, Theorem 3.1 and Corollary 3.6]. Theorem 2.3 Let R be a non-commutative prime ring, I an ideal of R, (F, d) a generalized derivation of R and a ∈ R. If a ∈ / Z and F ([x, a]) = 0 or [F (x), a] = 0 for all x ∈ I, then d(x) = λ[x, a], for all x ∈ I. Proof. Since a ∈ / Z and [F (x), a] = 0 for all x ∈ I, then by Lemma 2.1 we have F ([x, a]) = 0 and d(a) = 0 22 Neşet AYDIN By the proof of the Lemma 2.1, we have the equation (3); in the equation (3), replace y by [a, y] then we get 0 = (F (x)[a, y] + xd([a, y]) − xF ([a, y]))[r, a] + [x, a][a, y]d(r) = (F (x)[a, y] + x[a, d(y)][r, a] + [x, a][a, y]d(r) = −(F (x)[y, a] + x[d(y), a])[r, a] + [x, a][a, y]d(r) In the above equation, using the (10) equation [a, x]d(y) = F (x)[y, a]+x[d(y), a] in the proof of the Lemma 2.1, we obtain [a, x](d(y)[r, a] − [y, a]d(r)) = 0 Define h : R → R, h(x) = [a, x], then the above equation yields h(x)(d(y)[r, a]− [y, a]d(r)) = 0. Since a ∈ / Z, by [6, Lemma 1] we get d(y)[r, a] = [y, a]d(r), ∀y ∈ I, r ∈ R (15) Replace r by rs, s ∈ R in (15) and use (15), we obtain d(y)r[s, a] = [y, a]rd(s), ∀r, s ∈ R, y ∈ I (16) Substitute yz, z ∈ R instead of y in (16) and use (16) it gives us d(z)r[s, a] = [z, a]rd(s) (17) Now, define g : R → R, g(x) = [x, a], then from (17) we have d(z)rg(s) = g(z)rd(s), ∀r, s, z ∈ R Since g = 0, by [4, Lemma 1.3.2] for some λ ∈ C, d(x) = λ[x, a] Thus, the the proof is complete. References [1] Albas, E. and Argaç, N., Generalized Derivations of Prime Rings, Algebra Colloquium, 11:3, (2004), 399-410. [2] Bresar, M., On the distance of the composition of two derivations to the generalized derivations, Glasgow Math. J., 33, (1991), 89-93. Generalized Derivations of Prime Rings 23 [3] Havala, B., Generalized derivation in prime rings, Comm. Algebra, 26 (4), (1998), 1147-1166. [4] Herstein, I. N. Rings with Involution, Chicago Lectures in Mathematics, The University of Chicago Press, Chicago, Ill.-London, 1976. [5] Passman, D., Infinite Crossed Products, Academic Press, San Diego,1989. [6] Posner, E. C. Derivations in prime rings, Proc Amer. Math. Soc., 8, (1957), 1093-1100. Received: August, 2010