EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 11, No. 3, 2018, 717-729 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global Multiplicative (generalized) reverse derivations on semiprime ring Asma Ali1,∗, Ambreen Bano1 1 Department of Mathematics, Aligarh Muslim University, Aligarh, India Abstract. Let R be a semiprime ring. A mapping F : R→ R (not necessarily additive) is called a multiplicative (generalized) reverse derivation if there exists a map d : R→ R (not necessarily a derivation nor an additive map) such that F (xy) = F (y)x + yd(x) for all x, y ∈ R. In this paper we investigate some identities involving multiplicative (generalized) reverse derivation and prove some theorems in which we characterize these mappings. 2010 Mathematics Subject Classifications: 16U80, 16N60, 16W25 Key Words and Phrases: Semiprime ring, Ideal, Multiplicative (generalized) reverse derivation. 1. Introduction Let R be an associative ring. The centre of R is denoted by Z(R). For x, y ∈ R, the symbol [x, y] will denote the commutator xy − yx and the symbol x ◦ y will denote the anticommutator xy + yx. Recall that a ring R is prime if for any a, b ∈ R, aRb = 0 implies a = 0 or b = 0 and a ring R is semiprime if for any a ∈ R, aRa = 0 implies a = 0. An additive map d : R → R is called a derivation if d(xy) = d(x)y + xd(y) for all x, y ∈ R. The concept of derivation was extended to generalized derivation by Bresar [2]. An additive map F : R → R is said to be 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. Daif [4] introduced the concept of multiplicative derivation. A map D : R → R is said to be a multiplicative derivation if it satisfies D(xy) = D(x)y + xD(y) for all x, y ∈ R. Daif and Tammam El-Sayiad [5] extended the concept of multiplicative derivation to multiplicative generalized derivation. A mapping F : R → R is said to be a multiplicative generalized derivation if there exists a derivation d on R such that F (xy) = F (x)y + xd(y) for all x, y ∈ R. In this definition if we take d to be a mapping not necessarily a derivation nor an additive map, then F is said to be a multiplicative (generalized)-derivation which was introduced by Dhara and Ali [7]. Dhara and Ali [7] studied the following identities related ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v11i3.3248 Email addresses: asma ali2@rediffmail.com (Asma Ali), ambreenbn9@gmail.com (Ambreen Bano) http://www.ejpam.com 717 c© 2018 EJPAM All rights reserved. A. Ali, A. Bano / Eur. J. Pure Appl. Math, 11 (3) (2018), 717-729 718 on multiplicative (generalized) derivation on a semiprime ring: (i)F (xy) ± xy ∈ Z(R), (ii)F (xy)± yx ∈ Z(R), (iii)F (x)F (y)± xy ∈ Z(R) and (iv)F (x)F (y)± yx ∈ Z(R) for all x, y in some suitable subset of a semiprime ring R. The concept of reverse derivation was first time introduced by Herstein [8]. A mapping d : R → R which satisfies d(xy) = d(y)x + yd(x) for all x, y ∈ R is called a reverse derivation. Further Bresar and Vukman [3] studied the reverse derivation. Aboubakr and gonzalez [1] generalized the notion of reverse derivation by introducing generalized reverse derivation. An additive map F : R → R is said to be a generalized reverse derivation if F (xy) = F (y)x + yd(x) holds for all x, y ∈ R, where d is a reverse derivation of R. Very recently Tiwari et al [11] defined multiplicative (generalized) reverse derivation. A map F : R → R is said to be a multiplicative (generalized) reverse derivation if F (xy) = F (y)x + yd(x) holds for all x, y ∈ R, where d is any map on R. In the mentioned paper they proved commutativity of semiprime ring admitting a multiplicative (generalized) reverse derivation satisfying one of the following conditions: (i)F (x)F (y)± xy = 0, (ii)F (x)F (y)± yx = 0, (iii)F (xy)± [x, y] = 0, (iv)F (xy)± (x ◦ y) = 0, (v)F (xy)± F (x)F (y) = 0, for all x, y ∈ I, a two sided ideal in a semiprime ring R. As a line of investigation we study the following situations: (i)F (x)F (y)± xy ∈ Z(R) for all x, y ∈ I, (ii)F (x)F (y)± yx ∈ Z(R) for all x, y ∈ I, (iii)F (xy)± [x, y] ∈ Z(R) for all x, y ∈ I, (iv)F (xy)± x ◦ y ∈ Z(R) for all x, y ∈ I, (v)F (xy)± F (y)F (x) ∈ Z(R) for all x, y ∈ I, (vi)[F (x), y]± xy ∈ Z(R) for all x, y ∈ I, (vii)F (x) ◦ y ± xy ∈ Z(R) for all x, y ∈ I, (viii)[F (x), y]± yx ∈ Z(R) for all x, y ∈ I, (ix)F (x) ◦ y ± yx ∈ Z(R) for all x, y ∈ I. where I is a non zero ideal in a semiprime ring R. 2. Preliminaries Let R be a ring, we need the following basic identities which will be used in the proof of our results. For any x, y, z ∈ R, [xy, z] = x[y, z] + [x, z]y and [x, yz] = y[x, z] + [x, y]z. x ◦ (yz) = (x ◦ y)z − y[x, z] = y(x ◦ z) + [x, y]z. (xy) ◦ z = x(y ◦ z)− [x, z]y = (x ◦ z)y + x[y, z]. For any subset S of R, we will denote rR(S) the right annihilator of S in R, that is rR(S) = {x ∈ R|Sx = 0} and by lR(S) the left annihilator of S in R, that is, lR(S) = {x ∈ R|xS = 0}. If rR(S) = lR(S), then rR(S) is called an annihilator of R and is written as annR(S). For given x, y ∈ R, set [x, y]0 = x, [x, y]1 = [x, y] = xy − yx and [x, y]k = [[x, y]k−1, y] for k > 1. A. Ali, A. Bano / Eur. J. Pure Appl. Math, 11 (3) (2018), 717-729 719 Moreover, we shall require the following known results. Lemma 2.1. [9, Corollary 1] If R is a semiprime ring and I is an ideal of R, then rR(I) = lR(I). Lemma 2.2. [9, Corollary 2] If R is a semiprime ring and I is an ideal of R, then I ∩ annR(I) = 0. Lemma 2.3. [6, Fact-4] Let R be a semiprime ring, d a nonzero derivation of R such that x[[d(x), x], x] = 0 for all x ∈ R, then d maps R into its centre. 3. Main Results Theorem 3.1. Let R be a semiprime ring and F be a non-zero multiplicative (generalized) reverse derivation associated with a map d, I be a non-zero ideal of R. If F (x)F (y)±xy ∈ Z(R) for all x, y ∈ I, then [d(x), x] = 0 for all x ∈ I. Proof By the hypothesis, we have F (x)F (y) + xy ∈ Z(R) for all x, y ∈ I. (3.1) Substituting zy for y in (3.1) and using the definition of multiplicative (generalized) reverse derivation, we find F (x)F (y)z + F (x)yd(z) + xzy − xyz + xyz ∈ Z(R) for all x, y, z ∈ I. This implies that (F (x)F (y) + xy)z + F (x)yd(z) + x[z, y] ∈ Z(R). (3.2) Commuting (3.2) with z and using (3.1), we get [F (x)yd(z), z] + [x[z, y], z] = 0 for all x, y, z ∈ I. (3.3) Replacing x by zx in (3.3), we obtain [(F (x)z + xd(z))yd(z), z] + [zx[z, y], z] = 0. Therefore we get [F (x)zyd(z), z] + [xd(z)yd(z), z] + [zx[z, y], z] = 0. (3.4) Replacing y by zy in (3.3), we get [F (x)zyd(z), z] + [xz[z, y], z] = 0 for all x, y, z ∈ I. (3.5) A. Ali, A. Bano / Eur. J. Pure Appl. Math, 11 (3) (2018), 717-729 720 Subtracting (3.5) from (3.4), we obtain [xd(z)yd(z), z] + [[z, x][z, y], z] = 0. (3.6) Substituting yz for y in (3.6), we obtain [xd(z)yzd(z), z] + [[z, x][z, y], z]z = 0. (3.7) Right multiplying (3.6) by z and subtracting from (3.7), we get [xd(z)y[d(z), z], z] = 0 for all x, y, z ∈ I. (3.8) Replacing x by d(z)x in (3.8) and using (3.8), we find [d(z), z]xd(z)y[d(z), z] = 0 for all x, y, z ∈ I. (3.9) Substituting zy for y in (3.9), we obtain [d(z), z]xd(z)zy[d(z), z] = 0 for all x, y, z ∈ I. (3.10) Substituting xz for x in (3.9), we get [d(z), z]xzd(z)y[d(z), z] = 0 for all x, y, z ∈ I. (3.11) Subtracting (3.11) from (3.10), we have [d(z), z]x[d(z), z]y[d(z), z] = 0 for all x, y, z ∈ I. This implies that (I[d(z), z])3 = 0. Since a semiprime ring has no non-zero nilpotent left ideal, we get I[d(z), z] = 0 for all z ∈ I. Therefore, we get [d(z), z] ∈ I ∩ annR(I). By Lemma 2.2, we get [d(z), z] = 0 for all z ∈ I. By using similar argument, we arrive at the same conclusion for F (x)F (y)− xy ∈ Z(R) for all x, y ∈ I. Theorem 3.2. Let R be a semiprime ring and F be a non-zero multiplicative (generalized) reverse derivation associated with a map d, I be a non-zero ideal of R. If F (x)F (y)±yx ∈ Z(R) for all x, y ∈ I, then [d(x), x] = 0 for all x ∈ I. Proof By assumption, we have F (x)F (y) + yx ∈ Z(R) for all x, y ∈ I. (3.12) Replacing y by zy in (3.12), we get F (x)(F (y)z + yd(z)) + zyx + yxz − yxz ∈ Z(R) for all x, y, z ∈ I. This implies that (F (x)F (y) + yx)z + F (x)yd(z) + [z, yx] ∈ Z(R) (3.13) A. Ali, A. Bano / Eur. J. Pure Appl. Math, 11 (3) (2018), 717-729 721 Commuting (3.13) with z and using (3.12), we obtain [F (x)yd(z), z] + [[z, yx], z] = 0 for all x, y, z ∈ I. This implies that [F (x)yd(z), z] + [y[z, x], z] + [[z, y]x, z] = 0. (3.14) Replacing x by zx in (3.14), we obtain [F (x)zyd(z), z] + [xd(z)yd(z), z] + [yz[z, x], z] + [[z, y]zx, z] = 0. (3.15) Substituting zy for y in (3.14), we obtain [F (x)zyd(z), z] + [zy[z, x], z] + [z[z, y]x, z] = 0. (3.16) Subtracting (3.16) from (3.15), we get [xd(z)yd(z), z] + [[y, z][z, x], z] + [[[z, y], z]x, z] = 0. (3.17) Replacing x by xz in (3.17), we obtain [xzd(z)yd(z), z] + [[y, z][z, x], z]z + [[[z, y], z]x, z]z = 0 (3.18) Right multiplying (3.17) by z and subtracting from (3.18), we get [x[d(z)yd(z), z], z] = 0 for all x, y, z ∈ I. (3.19) This implies that [x, z][d(z)yd(z), z] + x[[d(z)yd(z), z], z] = 0. (3.20) Substituting ux for x in (3.20), we get u[x, z][d(z)yd(z), z] + [u, z]x[d(z)yd(z), z] + ux[[d(z)yd(z), z], z] = 0. for all x, y, z, u ∈ I. (3.21) Left multiplying (3.20) by u and subtracting from (3.21), we obtain [u, z]x[d(z)yd(z), z] = 0 for all x, y, z, u ∈ I. (3.22) Putting u = d(z)yd(z) in (3.22), we get (I[d(z)yd(z), z])2 = 0 for all y, z ∈ I. (3.23) Since a semiprime ring has no non-zero nilpotent left ideal, therefore we get I[d(z)yd(z), z] = 0, this implies that [d(z)yd(z), z)] ∈ I ∩ annR(I). By Lemma 2.2, we get [d(z)yd(z), z] = 0. This implies that d(z)yd(z)z − zd(z)yd(z) = 0 for all y, z ∈ I. (3.24) A. Ali, A. Bano / Eur. J. Pure Appl. Math, 11 (3) (2018), 717-729 722 Replacing y by yd(z)w in (3.24), we get d(z)yd(z)wd(z)z − zd(z)yd(z)wd(z) = 0 for all y, z, w ∈ I. (3.25) Using (3.24), (3.25) gives d(z)yzd(z)wd(z)− d(z)yd(z)zwd(z) = 0. This implies that d(z)y[d(z), z]wd(z) = 0 for all y, z, w ∈ I. This yields that [d(z), z]y[d(z), z]w[d(z), z] = 0 for all y, z, w ∈ I. That is (I[d(z), z])3 = 0 for all z ∈ I. Since a semiprime ring has no non-zero nilpotent left ideal, we get I[d(z), z] = 0 which implies that [d(z), z] ∈ I∩annR(I). By Lemma 2.2, we get [d(z), z] = 0. Similarly we can prove the result for the case F (x)F (y)− yx ∈ Z(R) for all x, y ∈ I. Theorem 3.3. Let R be a semiprime ring and F be a non-zero multiplicative (generalized) reverse derivation associated with a map d, I be a non-zero ideal of R. If F (xy)± [x, y] ∈ Z(R) for all x, y ∈ I, then x[d(x), x]2 = 0 for all x ∈ I. Proof By the hypothesis, we have F (xy) + [x, y] ∈ Z(R) for all x, y ∈ I. (3.26) Replacing x by zx in (3.26), we get F (xy)z + xyd(z) + z[x, y] + [z, y]x ∈ Z(R) for all x, y, z ∈ I. This implies that (F (xy) + [x, y])z + xyd(z) + [z, [x, y]] + [z, y]x ∈ Z(R). (3.27) Commuting (3.27) with z and using (3.26), we get [xyd(z), z] + [[z, [x, y]], z] + [[z, y]x, z] = 0 for all x, y, z ∈ I. (3.28) Replacing y by zy in (3.28), we get [xzyd(z), z] + [[z, [x, zy]], z] + [[z, zy]x, z] = 0. [xzyd(z), z] + [[z, z[x, y] + [x, z]y], z] + [z[z, y]x, z] = 0. [xzyd(z), z] + [[z, z[x, y]], z] + [[z, [x, z]y], z] + z[[z, y]x, z] = 0. This implies that [xzyd(z), z] + z[[z, [x, y]], z] + [[x, z][z, y], z] + [[z, [x, z]]y, z] + z[[z, y]x, z] = 0. (3.29) Left multiplying (3.28) by z and subtracting from (3.29), we obtain [[z, x]yd(z), z]− [[x, z][z, y], z]− [[z, [x, z]]y, z] = 0 for all x, y, z ∈ I. (3.30) A. Ali, A. Bano / Eur. J. Pure Appl. Math, 11 (3) (2018), 717-729 723 Replacing y by yz in (3.30), we get [[z, x]yzd(z), z]− [[x, z][z, y], z]z − [[z, [x, z]]y, z]z = 0. (3.31) Right multiplying (3.30) by z and subtracting from (3.31), we obtain [[z, x]y[d(z), z], z] = 0 for all x, y, z ∈ I. (3.32) Substituting zd(z) for x in (3.32), we get [z[z, d(z)]y[d(z), z], z] = 0 for all y, z ∈ I. (3.33) Replacing y by yz in (3.33), we obtain [z[d(z), z]yz[d(z), z], z] = 0 for all y, z ∈ I. This implies that z[d(z), z]yz[d(z), z]z − z2[d(z), z]yz[d(z), z] = 0. (3.34) Replacing y by yz[d(z), z]w in (3.34), we get z[d(z), z]yz[d(z), z]wz[d(z), z]z − z2[d(z), z]yz[d(z), z]wz[d(z), z] = 0 for all y, z, w ∈ I. (3.35) Using (3.34) in (3.35), (3.35) yields that z[d(z), z]yz2[d(z), z]wz[d(z), z]− z[d(z), z]yz[d(z), z]zwz[d(z), z] = 0. This implies that z[d(z), z]y[z[d(z), z], z]wz[d(z), z] = 0 for all y, z, w ∈ I. After a simple calculation this gives that [z[d(z), z], z]y[z[d(z), z], z]w[z[d(z), z], z] = 0. (3.36) This implies that (I[z[d(z), z], z])3 = 0 for all z ∈ I. Since a semiprime ring has no non-zero nilpotent left ideal, we get I[z[d(z), z], z] = 0 for all z ∈ I. This implies that [z[d(z), z], z] ∈ I ∩ annR(I). By Lemma 2.2, we get that [z[d(z), z], z] = 0 that is z[d(z), z]2 = 0 for all z ∈ I. By using similar argument we can get the result for the case F (xy)− [x, y] ∈ Z(R) for all x, y ∈ I. Theorem 3.4. Let R be a semiprime ring and F be a non-zero multiplicative (generalized) reverse derivation associated with a map d, I be a non-zero ideal of R. If F (xy)± x ◦ y ∈ Z(R) for all x, y ∈ I, then x[d(x), x]2 = 0 for all x ∈ I. A. Ali, A. Bano / Eur. J. Pure Appl. Math, 11 (3) (2018), 717-729 724 Proof By the hypothesis, we have F (xy) + x ◦ y ∈ Z(R) for all x, y ∈ I. (3.37) Replacing x by zx in (3.37), we get F (xy)z + xyd(z) + z(x ◦ y)− [z, y]x + (x ◦ y)z − (x ◦ y)z ∈ Z(R) for all x, y, z ∈ I. This implies that (F (xy) + x ◦ y)z + xyd(z) + [z, x ◦ y]− [z, y]x ∈ Z(R). (3.38) Commuting (3.38) with z and using (3.37) we get [xyd(z), z] + [[z, x ◦ y], z]− [[z, y]x, z] = 0 for all x, y, z ∈ I. (3.39) Substituting zy for y in (3.39), we obtain [xzyd(z), z] + [[z, x ◦ zy], z]− [z[z, y]x, z] = 0. This implies that [xzyd(z), z] + [[z, z(x ◦ y) + [x, z]y], z]− [z[z, y]x, z] = 0. A simple calculation yields that [xzyd(z), z] + z[[z, x ◦ y], z] + [[x, z][z, y], z] + [[z, [x, z]]y, z]− z[[z, y]x, z] = 0. (3.40) Left multiplying (3.39) by z and subtracting from (3.40), we obtain [[z, x]yd(z), z]− [[x, z][z, y], z]− [[z, [x, z]]y, z] = 0. (3.41) This is the same as (3.30) in Theorem 3.3. Then by the same argument as we have used in Theorem 3.3, we get the result. In the similar manner the conclusion can be obtained for the case F (xy)− x ◦ y ∈ Z(R) for all x, y ∈ I. Theorem 3.5. Let R be a semiprime ring and F be a non-zero multiplicative (gen- eralized) reverse derivation associated with a map d, I be a non-zero ideal of R. If F (xy)± F (y)F (x) ∈ Z(R) for all x, y ∈ I, then x[d(x), x]2 = 0 for all x ∈ I. Proof By the hypothesis, we have F (xy) + F (y)F (x) ∈ Z(R) for all x, y ∈ I. (3.42) Replacing x by zx in (3.42), we get F (xy)z + xyd(z) + F (y)F (x)z + F (y)xd(z) ∈ Z(R) for all x, y, z ∈ I. (3.43) A. Ali, A. Bano / Eur. J. Pure Appl. Math, 11 (3) (2018), 717-729 725 Commuting (3.43) with z and using (3.42), we have [xyd(z), z] + [F (y)xd(z), z] = 0 for all x, y, z ∈ I. (3.44) Replacing y by z2 and applying definition of F in (3.44), we get [xz2d(z), z] + [F (z)zxd(z), z] + [zd(z)xd(z), z] = 0. (3.45) Replacing x by zx and y by z in (3.44), we get [zxzd(z), z] + [F (z)zxd(z), z] = 0 for all x, z ∈ I. (3.46) Subtracting (3.46) from (3.45), we obtain [[x, z]zd(z), z] + [zd(z)xd(z), z] = 0 for all x, z ∈ I. (3.47) Substituting zx for x in (3.47), we get z[[x, z]zd(z), z] + [zd(z)zxd(z), z] = 0. (3.48) Left multiplying (3.47) by z and subtracting from (3.48), we obtain [z[d(z), z]xd(z), z] = 0 for all x, z ∈ I. (3.49) Substituting xz for x in (3.49), we get [z[d(z), z]xzd(z), z] = 0 for all x, z ∈ I. (3.50) Right multiplying (3.49) by z and subtracting from (3.50), we get [z[d(z), z]x[d(z), z], z] = 0 for all x, z ∈ I. (3.51) This is the same as (3.33) in Theorem 3.3. Then by using the same technique as we have used in Theorem 3.3, we get the result. In a similar manner we can prove the result for the case F (xy)− F (y)F (x) ∈ Z(R) for all x, y ∈ I. An immediate consequence of above Theorems and together with Lemma 2.3 we have the following corollary: Corollary 3.1. Let R be a semiprime ring and F be a non-zero multiplicative (general- ized) reverse derivation associated with a non-zero derivation d : R→ R. Then d maps R into Z(R), the centre of R if for all x, y ∈ R, one of the following holds: (i)F (xy)± [x, y] ∈ Z(R). (ii)F (xy)± x ◦ y ∈ Z(R). (iii)F (xy)± F (y)F (x) ∈ Z(R). A. Ali, A. Bano / Eur. J. Pure Appl. Math, 11 (3) (2018), 717-729 726 Theorem 3.6. Let R be a semiprime ring and F be a non-zero multiplicative (generalized) reverse derivation associated with a map d, I be a non-zero ideal of R. If [F (x), y]± xy ∈ Z(R) for all x, y ∈ I, then [d(x), x] = 0 for all x ∈ I. Proof By the hypothesis, we have [F (x), y] + xy ∈ Z(R) for all x, y ∈ I. (3.52) Replacing x by zx in (3.52), we get [F (x)z + xd(z), y] + zxy + xyz − xyz ∈ Z(R) for all x, y, z ∈ I. This gives that F (x)[z, y] + [F (x), y]z + x[d(z), y] + [x, y]d(z) + xyz + [z, xy] ∈ Z(R). (3.53) Commuting (3.53) with z and using (3.52), we obtain [F (x)[z, y], z] + [x[d(z), y], z] + [[x, y]d(z), z] + [x[z, y] + [z, x]y, z] = 0. This implies that [F (x)[z, y], z] + [x[d(z), y], z] + [[x, y]d(z), z] + [x[z, y], z] + [[z, x]y, z] = 0. (3.54) Substituting yz for y in (3.54), we get [F (x)[z, y], z]z + [xy[d(z), z], z] + [x[d(z), y], z]z + [y[x, z]d(z), z] +[[x, y]zd(z), z] + [x[z, y], z]z + [[z, x]y, z]z = 0. (3.55) Right multiplying (3.54) by z and subtracting from (3.55), we obtain [[x, y]d(z)z, z]− [xy[d(z), z], z]− [y[x, z]d(z), z]− [[x, y]zd(z), z] = 0. (3.56) This implies that [xyd(z)z−yxd(z)z−xyd(z)z+xyzd(z), z]−[yxzd(z)−yzxd(z), z]−[xyzd(z)−yxzd(z), z] = 0. After a simple calculation this yields that [y[xd(z), z], z] = 0 for all x, y, z ∈ I. (3.57) Replacing x by d(z)x in (3.57), we get [y[d(z)xd(z), z], z] = 0 for all x, y, z ∈ I. (3.58) This is the same as (3.19) in Theorem 3.2. We can complete the proof by using similar technique as we have used in Theorem 3.2. In the similar manner the conclusion can be obtained for [F (x), y]− xy ∈ Z(R) for all x, y ∈ I. Using similar technique with some necessary variations, we can prove the following: A. Ali, A. Bano / Eur. J. Pure Appl. Math, 11 (3) (2018), 717-729 727 Theorem 3.7. Let R be a semiprime ring and F be a non-zero multiplicative (generalized) reverse derivation associated with a map d, I be a non-zero ideal of R. If F (x) ◦ y± xy ∈ Z(R) for all x, y ∈ I, then [d(x), x] = 0 for all x ∈ I. Theorem 3.8. Let R be a semiprime ring and F be a non-zero multiplicative (generalized) reverse derivation associated with a map d, I be a non-zero ideal of R. If [F (x), y]± yx ∈ Z(R) for all x, y ∈ I, then [F (x), x]2x = 0 and x[d(x), x]2 = 0 for all x ∈ I. Proof We begin with the situation [F (x), y] + yx ∈ Z(R) for all x, y ∈ I. (3.59) Replacing y by yz in (3.59), we get y[F (x), z] + [F (x), y]z + yzx + yxz − yxz ∈ Z(R) for all x, y, z ∈ I. This implies that ([F (x), y] + yx)z + y[F (x), z] + y[z, x] ∈ Z(R). (3.60) Commuting (3.60) with z and applying (3.59), we obtain [y[F (x), z], z] + [y[z, x], z] = 0 for all x, y, z ∈ I. (3.61) Substituting zx for x in (3.61), we obtain [y[F (x)z + xd(z), z], z] + [yz[z, x], z] = 0 for all x, y, z ∈ I. This gives that [y[F (x), z], z]z + [y[xd(z), z], z] + [yz[z, x], z] = 0. (3.62) Replacing y by ry in (3.62), we obtain r[y[F (x), z], z]z + [r, z]y[F (x), z]z + r[y[xd(z), z], z] +[r, z]y[xd(z), z] + r[yz[z, x], z] + [r, z]yz[z, x] = 0. (3.63) Left multiplying (3.62) by r and subtracting from (3.63), we get [r, z]y{[F (x), z]z + [xd(z), z] + z[z, x]} = 0 for all x, y, z ∈ I and r ∈ R. (3.64) In particular take x = z in (3.64), we obtain [r, z]y{[F (z), z]z + z[d(z), z]} = 0 for all y, z ∈ I and r ∈ R. This implies that [r, z]y[F (z)z + zd(z), z] = 0 for all y, z ∈ I and r ∈ R. REFERENCES 728 Since F is a multiplicative (generalized) reverse derivation, we conclude that [r, z]y[F (z2), z] = 0 for all y, z ∈ I and r ∈ R. (3.65) Putting r = F (z2) and using semiprimeness of R, we have [F (z2), z] = 0 for all z ∈ I. Replacing yx in place of x in (3.59), we have [F (x)y + xd(y), y] + y2x ∈ Z(R) for all x, y ∈ I. That is ([F (x), y] + yx)y + [xd(y), y] + y[y, x] ∈ Z(R) for all x, y ∈ I. (3.66) Commuting (3.66) with y and using (3.59), we have [[xd(y), y], y] + y[[y, x], y] = 0 for all x, y ∈ I. (3.67) In particular for y = x in (3.67), we have x[d(x), x]2 = 0 for all x ∈ I. Since [F (x2), x] = 0. It implies that [F (x), x]x + x[d(x), x] = 0 for all x ∈ I. (3.68) This can be written as [F (x), x]2x+x[d(x), x]2 = 0 for all x ∈ I. By using x[d(x), x]2 = 0. This implies that [F (x), x]2x = 0 for all x ∈ I, which is our desired result. In the similar manner the conclusion can be obtained for the case [F (x), y]− yx ∈ Z(R) for all x, y ∈ I. Using similar technique with some necessary variations, we can prove the following: Theorem 3.9. Let R be a semiprime ring and F be a non-zero multiplicative (generalized) reverse derivation associated with a map d, I be a non-zero ideal of R. If F (x) ◦ y± yx ∈ Z(R) for all x, y ∈ I, then [F (x), x]2x = 0 and x[d(x), x]2 = 0 for all x ∈ I. ACKNOWLEDGEMENTS The authors are thankful to the referee for the valuable comments and suggestions. References [1] Aboubakr, A. and Gonzalez, S., Generalized reverse derivations on semiprime rings, Siberian Math. J. 56(2) (2015), 199-205. [2] Bresar, M., On the distance of the composition of two derivations to the generalized derivations, Glasgow Math. J. 33(1) (1991), 89-93. [3] Bresar, M. and Vukman, J., On some additive mappings in rings with involution, Aequat. Math. 38 (1989), 178-185. REFERENCES 729 [4] M. N. Daif, When is a multiplicative derivation additive, Int. J. Math. Math. Sci. 14(3) (1991), 615-618. [5] Daif, M.N., and Tammam El-Sayiad, M.S., Multiplicative generalized derivations which are additive, East-west J. Math. 9(1) (1997), 31-37. [6] Dhara, B. and Ali, S., On n-centralizing generalized derivations in semiprime rings with applications to C∗-algebras, J. Algebra and its applications 11(6) (2012) paper no-1250111, (11 Pages). [7] Dhara, B. and Ali, S., On multiplicative (generalized)-derivations in prime and semiprime rings, Aequat. Math. 86 (1-2) (2013), 65-79. [8] Herstein, I.N., Jordan derivations of prime rings, Proc. Amer. Math. Soc. 8 (1957), 1104-1110. [9] Herstein, I. N., Rings with involution, Chicago lectures in mathematics, University of Chicago press, Chicago III USA (1976). [10] Koc, E., Some results in semiprime rings with derivation, Commun. Fac. Sci. Univ. Ank. Series 62(1) (2013), 11-20. [11] Tiwari, S.K., Sharma, R.K. and Dhara, B., Some theorems of commutativity on semiprime rings with mappings, Southeast Asian Bull. Math. 42 (2018), 579-592.