� EXTRACTA MATHEMATICAE Volumen 33, Número 2, 2018 instituto de investigación de matemáticas de la universidad de extremadura EXTRACTA MATHEMATICAE Vol. 38, Num. 1 (2023), 67 – 84 doi:10.17398/2605-5686.38.1.67 Available online May 3, 2023 On Lie ideals satisfying certain differential identities in prime rings B. Dhara 1, S. Ghosh 2,∗, G.S. Sandhu 3 1 Department of Mathematics, Belda College, Belda Paschim Medinipur, 721424, W.B., India 2 Department of Mathematics, Jadavpur University, Kolkata-700032, W.B., India 3 Department of Mathematics, Patel Memorial National College 140401 Rajpura, India basu dhara@yahoo.com , mathsourav99@gmail.com , gurninder rs@pbi.ac.in Received January 20, 2023 Presented by C. Mart́ınez Accepted March 28, 2023 Abstract: Let R be a prime ring of characteristic not 2, L a nonzero square closed Lie ideal of R and let F : R → R, G : R → R be generalized derivations associated with derivations d : R → R, g : R → R respectively. In this paper, we study several conditions that imply that the Lie ideal is central. Moreover, it is shown that the assumption of primeness of R can not be removed. Key words: Prime ring, derivation, generalized derivation, Lie ideal. MSC (2020): 16N60, 16W25, 16R50. 1. Introduction Throughout this paper R denotes an associative prime ring with center Z(R). A ring R is called a prime ring if for any a, b ∈ R, aRb = (0) implies either a = 0 or b = 0. The symbol [x, y]=xy − yx stands for the commutator operator for x, y ∈ R and the symbol x ◦ y = xy + yx stands for the anti- commutator operator for x, y ∈ R. An additive subgroup L of R is said to be a Lie ideal of R if [x, r] ∈ L for any r ∈ R and x ∈ L. A Lie ideal L is said to be square closed if u2 ∈ L for all u ∈ L. A map d : R → R is called a derivation, if d(x + y) = d(x) + d(y) and d(xy) = d(x)y + xd(y) holds for all x, y ∈ R. An additive map F : R → R is said to be a generalized derivation of R, if there exists a derivation d : R → R such that F (xy) = F (x)y + xd(y) holds for all x, y ∈ R. Obviously, every ∗The second author expresses his thanks to the University Grants Commission, New Delhi for its JRF awarded to him. The grant No. is UGC-Ref.No.1156/(CSIR-UGC NET DEC. 2018) dated 24.07.2019. ISSN: 0213-8743 (print), 2605-5686 (online) c© The author(s) - Released under a Creative Commons Attribution License (CC BY-NC 3.0) https://doi.org/10.17398/2605-5686.38.1.67 mailto:basu_dhara@yahoo.com mailto:mathsourav99@gmail.com mailto:gurninder_rs@pbi.ac.in https://publicaciones.unex.es/index.php/EM https://creativecommons.org/licenses/by-nc/3.0/ 68 b. dhara, s. ghosh, g.s. sandhu derivation is a generalized derivation of R, but the converse is not necessarily true. Several authors have studied many identities in prime and semiprime rings, involving derivations and generalized derivations, that imply commutativity of the ring. We refer the reader to [1, 2, 7, 8, 9, 11, 12, 15, 16, 18], where further references can be found. The identities (i) F (xy) −xy ∈ Z(R) for all x, y ∈ I, (ii) F (xy) + xy ∈ Z(R) for all x, y ∈ I, (iii) F (xy) −yx ∈ Z(R) for all x, y ∈ I, (iv) F (xy) + yx ∈ Z(R) for all x, y ∈ I, (v) F (x)F (y) −xy ∈ Z(R) for all x, y ∈ I, (vi) F (x)F (y) + xy ∈ Z(R) for all x, y ∈ I, when R is a prime ring, F is a generalized derivation of R associated with a non-zero derivation d and I is a non-zero two-sided ideal of R, were studied by Ashraf et al. in [2], proving that any of them implies the commu- tativity of R. In a similar way, the identities (i) F (x)F (y)−yx ∈ Z(R) and (ii) F (x)F (y)+ yx ∈ Z(R) for all x, y in some suitable subset of R were studied in [9] by Dhara et al. Recently, Tiwari et al. (see [18]) studied identities involving three sum- mands and again obtained the commutativity of the prime ring R. Identities involving the commutator and the anti-commutator have also been considered. In [4], Bell and Daif proved that if U is a nonzero right ideal of a semiprime ring R and d is a nonzero derivation of R such that [d(x), d(y)] = [x, y] for all x, y ∈ U, then U ⊆ Z(R). Ashraf et al. (see [3]) got the commutativity of a prime ring R satisfying any one of the following conditions: (i) d(x) ◦F (y) = 0 for all x, y ∈ I, (ii) [d(x), F (y)] = 0 for all x, y ∈ I, (iii) d(x) ◦F (y) = x◦y for all x, y ∈ I, (iv) d(x) ◦F (y) + x◦y = 0 for all x, y ∈ I, (v) [d(x), F (y)] = [x, y] for all x, y ∈ I, (vi) [d(x), F (y)] + [x, y] = 0 for all x, y ∈ I, (vii) d(x)F (y) ±xy ∈ Z(R) for all x, y ∈ I, on lie ideals satisfying certain differential identities 69 where I is a nonzero ideal of R and F is a generalized derivation of R associated with a nonzero derivation d. In [13], Shuliang studied the above identities for a square closed Lie ideal L of a prime ring R and obtained that either d = 0 or L ⊆ Z(R) and in [10], Dhara et al. studied the above identities in a semiprime ring. In this paper we consider the following identities: (i) F (u) ◦v ±d(u) ◦F (v) ±u◦v = 0 for all u, v ∈ L, (ii) [F (u), v] ± [d(u), F (v)] ± [u, v] = 0 for all u, v ∈ L, (iii) F ([u, v]) ± [d(u), F (v)] ± [u, v] = 0 for all u, v ∈ L, (iv) F (u◦v) ± [d(u), F (v)] ±u◦v = 0 for all u, v ∈ L, (v) F (u)G(v) ±d(u)F (v) ±uv ∈ Z(R) for all u, v ∈ L, (vi) G(uv) ±d(u)F (v) ±F (vu) = 0 for all u, v ∈ L, (vii) F (uv) ±F (v)F (u) ±uv ∈ Z(R) for all u, v ∈ L, where L is a square closed Lie ideal in a prime ring R and F is a generalized derivation of R associated with a derivation d. 2. Preliminaries In this paper R will denote always a prime associative ring of characteristic not equal to 2. This implies that for any element x in R, 2x = 0 implies x = 0. Let L be a square closed Lie ideal of R. Thus u2 ∈ L for all u ∈ L. Now for u, v ∈ L, uv + vu = (u + v)2 − u2 − v2 ∈ L and by definition of Lie ideal uv −vu ∈ L. Combining these two we get 2uv ∈ L for all u, v ∈ L. The following identities shall be used very frequently throughout: (1) [xy, z] = x[y, z] + [x, z]y for all x, y, z ∈ R, (2) [x, yz] = y[x, z] + [x, y]z for all x, y, z ∈ R, (3) (x◦yz) = (x◦y)z −y[x, z] = y(x◦z) + [x, y]z for all x, y, z ∈ R, (4) (xy ◦z) = x(y ◦z) − [x, z]y = (x◦z)y + x[y, z] for all x, y, z ∈ R. Moreover in order to prove our results, we need the following facts: Lemma 2.1. ([5, Lemma 2]) If L 6⊆ Z(R) is a Lie deal of R, then CR(L) = Z(R). Lemma 2.2. ([5, Lemma 4]) If L 6⊆ Z(R) is a Lie ideal of R and aLb = 0, then either a = 0 or b = 0. 70 b. dhara, s. ghosh, g.s. sandhu Lemma 2.3. ([5, Lemma 5]) If d is a nonzero derivation of R and L a Lie ideal of R such that d(L) = (0), then L ⊆ Z(R). Lemma 2.4. ([9, Lemma 2.5]) Let L be a nonzero Lie ideal of R and V = {u ∈ L | d(u) ∈ L}. Then V is also a nonzero Lie ideal of R. Moreover, if L is noncentral, then V is also noncentral. Lemma 2.5. ([14, Theorem 5]) Let d be a nonzero derivation of R and L a nonzero Lie ideal of R such that [u, d(u)] ∈ Z(R) for all u ∈ L. Then L ⊆ Z(R). Lemma 2.6. ([17, Lemma 2.6]) Let L 6⊆ Z(R) be a Lie ideal of R and a, b ∈ R such that one of a, b is in L. If aub + bua = 0 for all u ∈ L, then aub = bua = 0 for all u ∈ L. Consequently, a = 0 or b = 0. Lemma 2.7. Let d be a nonzero derivation of R and L a nonzero square closed Lie ideal of R. Suppose that V = {u ∈ L | d(u) ∈ L}. If [d(u), d(v)] = 0 for all u ∈ L and v ∈ V , then L ⊆ Z(R). Proof. If L ⊆ Z(R), we are done. Thus on contrary, we assume that L 6⊆ Z(R). By Lemma 2.4, V is also noncentral Lie ideal of R such that V ⊆ L. We have that [d(u), d(v)] = 0 (1) for all u ∈ L and v ∈ V . Replacing u by 2uv and then using char (R) 6= 2, we have [d(u)v + ud(v), d(v)] = 0 (2) for all u ∈ L and v ∈ V . By using (1), we have d(u)[v, d(v)] + [u, d(v)]d(v) = 0 (3) for all u ∈ L and v ∈ V . Replacing u by 2wu in (3) and then using it, we have d(w)u[v, d(v)] + [w, d(v)]ud(v) = 0 (4) for all u, w ∈ L and v ∈ V . In particular, d(v)u[v, d(v)] + [v, d(v)]ud(v) = 0 (5) on lie ideals satisfying certain differential identities 71 for all u ∈ L and v ∈ V . Invoking Lemma 2.6, we find d(v)u[v, d(v)] = 0 for all v ∈ V and u ∈ U. Again by Lemma 2.2, d(v) = 0 or [v, d(v)] = 0. In any case we have [v, d(v)] = 0 for all v ∈ V . By Lemma 2.5, V ⊆ Z(R), a contradiction. The following lemmas are the particular cases of [6, Theorem 1]. Lemma 2.8. Let d be a nonzero derivation of R and L a nonzero Lie ideal of R such that u[[d(u), u], u] = 0 for all u ∈ L. Then L ⊆ Z(R). Lemma 2.9. Let F be a nonzero generalized derivation of R with associ- ated nonzero derivation d and L a nonzero square closed Lie ideal of R such that [F (u), u] = 0 for all u ∈ L. Then L ⊆ Z(R). 3. Main results Theorem 3.1. Let L be a nonzero square closed Lie ideal of R and F be a generalized derivation of R associated to nonzero derivation d of R. If F (u) ◦v ±d(u) ◦F (v) ±u◦v = 0 for all u, v ∈ L, then L ⊆ Z(R). Proof. We assume on the contrary that L 6⊆ Z(R). By hypothesis, we have F (u) ◦v ±d(u) ◦F (v) ±u◦v = 0 (6) for all u, v ∈ L. Replacing v by 2vw in (6), we obtain 2{F (u) ◦vw ±d(u) ◦ (F (v)w + vd(w)) ±u◦ (vw)} = 0 (7) for all u, v, w ∈ L. Since characteristic of R is not 2, we have (F (u) ◦v)w −v[F (u), w] ± (d(u) ◦F (v))w ∓F (v)[d(u), w] ±d(u) ◦ (vd(w)) ± (u◦v)w ∓v[u, w] = 0 (8) for all u, v, w ∈ L. Right multiplying (6) by w and then subtracting from (8), we get −v[F (u), w] ∓F (v)[d(u), w] ±d(u) ◦ (vd(w)) ∓v[u, w] = 0 (9) for all u, v, w ∈ L. 72 b. dhara, s. ghosh, g.s. sandhu Substituting v by 2pv in (9) and then using characteristic of R is not 2, we obtain −pv[F (u), w] ∓F (pv)[d(u), w] ±p(d(u) ◦ (vd(w))) ±[d(u), p]vd(w) ∓pv[u, w] = 0 (10) for all u, v, w, p ∈ L. Left multiplying (9) by p and then subtracting from (10), we get ∓(F (pv) −pF (v))[d(u), w] ± [d(u), p]vd(w) = 0 (11) for all u, v, w, p ∈ L. Replacing w by 2wt in (11) and then using characteristic of R is not 2 and (11), we obtain ∓(F (pv) −pF (v))w[d(u), t] ± [d(u), p]vwd(t) = 0 (12) for all u, v, w, p, t ∈ L. Since [q, d(s)] ∈ L, thus 2[q, d(s)]w ∈ L for all q, s, w ∈ L. Replacing w by 2[q, d(s)]w in (12) and then using characteristic of R is not 2, we get ∓(F (pv) −pF (v))[q, d(s)]w[d(u), t] ± [d(u), p]v[q, d(s)]wd(t) = 0 (13) for all u, v, w, p, q, s, t ∈ L. Now re-writing the relation (11), we can write ∓(F (pv) −pF (v))[q, d(s)] ± [p, d(s)]vd(q) = 0 (14) for all v, p, q, s ∈ L. By using (14) in (13), we have ∓[p, d(s)]vd(q)w[d(u), t] ± [d(u), p]v[q, d(s)]wd(t) = 0 (15) for all u, v, w, p, q, s, t ∈ L. We re-write it as ∓[d(s), p]vd(q)w[d(u), t] ± [d(u), p]v[d(s), q]wd(t) = 0 (16) for all u, v, w, p, q, s, t ∈ L. Let V = {u ∈ L | d(u) ∈ L}. By Lemma 2.4, V is also noncentral Lie ideal of R such that V ⊆ L. Thus if s ∈ V , then d(s) ∈ L. Thus replacing p by 2d(s)p in (16) and then using characteristic of R is not 2 and the relation (16), we get ±[d(u), d(s)]pv[d(s), q]wd(t) = 0 (17) on lie ideals satisfying certain differential identities 73 for all u, v, w, p, q, t ∈ L and s ∈ V . By Lemma 2.2, for each s ∈ V , either [d(u), d(s)] = 0 for all u ∈ L or [d(s), q]wd(t) = 0 for all w ∈ L. Now we consider two subgroups T1 = { s ∈ V : [d(u), d(s)] = 0 for all u ∈ L } , T2 = { s ∈ V : [d(s), q]wd(t) = 0 for all q, w, t ∈ L } . It is very clear that T1 and T2 are two additive subgroups of V such that T1 ∪T2 = V . Since an additive subgroups can not be union of its two proper subgroups, we have either T1 = V or T2 = V , that is, either [d(u), d(s)] = 0 for all u ∈ L and s ∈ V or [d(s), q]wd(t) = 0 for all q, w, t ∈ L and s ∈ V . If [d(u), d(s)] = 0 for all u, s ∈ L, then by Lemma 2.7, L ⊆ Z(R), a contradiction. On the other hand, if [d(s), q]wd(t) = 0 for all q, w, t ∈ L and s ∈ V , then [d(s), q]w[d(s), q] = 0 for all w ∈ L and q, s ∈ V . By Lemma 2.2, [d(s), q] = 0 for all q, s ∈ V . By Lemma 2.5, V ⊆ Z(R), a contradiction. Thus the proof of the theorem is completed. Theorem 3.2. Let L be a nonzero square closed Lie ideal of R and F be a generalized derivation of R associated to nonzero derivation d of R. If [F (u), v] ± [d(u), F (v)] ± [u, v] = 0 for all u, v ∈ L, then L ⊆ Z(R). Proof. We assume on the contrary that L 6⊆ Z(R). By hypothesis, we have [F (u), v] ± [d(u), F (v)] ± [u, v] = 0 (18) for all u, v ∈ L. Replacing v by 2vw in (18), we obtain 2{[F (u), vw] ± [d(u), F (v)w + vd(w)] ± [u, vw]} = 0 (19) for all u, v, w ∈ L. Since characteristic of R is not 2, we have [F (u), v]w + v[F (u), w] ± ([d(u), F (v)]w + F (v)[d(u), w]) ±[d(u), vd(w)] ± ([u, v]w + v[u, w]) = 0 (20) for all u, v, w ∈ L. Right multiplying (18) by w and then subtracting from (20), we get v[F (u), w] ±F (v)[d(u), w]) ± [d(u), vd(w)] ±v[u, w] = 0 (21) 74 b. dhara, s. ghosh, g.s. sandhu for all u, v, w ∈ L. Replacing v by 2pv in (21) and then usingcharacteristic of R is not 2, we obtain pv[F (u), w] ±F (pv)[d(u), w]) ±p[d(u), vd(w)] ±[d(u), p]vd(w) ±pv[u, w] = 0 (22) for all u, v, w, p ∈ L. Left multiplying (21) by p and then subtracting from (22), we get ±(F (pv) −pF (v))[d(u), w]) ± [d(u), p]vd(w) = 0 (23) for all u, v, w, p ∈ L. Using the same arguments used in the proof of Theorem 3.1 we get a contradiction, which proves this theorem. Theorem 3.3. Let L be a nonzero square closed Lie ideal of R and F be a generalized derivation of R associated to nonzero derivation d of R. If F ([u, v]) ± [d(u), F (v)] ± [u, v] = 0 for all u, v ∈ L, then L ⊆ Z(R). Proof. We assume on the contrary that L 6⊆ Z(R). By hypothesis, we have F ([u, v]) ± [d(u), F (v)] ± [u, v] = 0 (24) for all u, v ∈ L. Replacing u by 2uv in (24), we obtain 2(F ([u, v])v + [u, v]d(v) ± [d(u)v, F (v)] ± [ud(v), F (v)] ± [u, v]v) = 0 (25) for all u, v ∈ L. Since characteristic of R is not 2, we have F ([u, v])v + [u, v]d(v) ± [d(u)v, F (v)] ± [ud(v), F (v)] ± [u, v]v = 0 (26) for all u, v ∈ L. Right multiplying (24) by v and then subtracting from (26), we get [u, v]d(v) ±d(u)[v, F (v)] ± [ud(v), F (v)] = 0 (27) for all u, v ∈ L. Replacing u by 2vu in (27) and then using characteristic of R is not 2, we obtain v[u, v]d(v) ± (d(v)u + vd(u))[v, F (v)] ± [vud(v), F (v)] = 0 on lie ideals satisfying certain differential identities 75 that is, v[u, v]d(v) ±d(v)u[v, F (v)] ±vd(u)[v, F (v)] ±v[ud(v), F (v)] ± [v, F (v)]ud(v) = 0 (28) for all u, v ∈ L. Left multiplying (27) by v and then subtracting from (28), we get d(v)u[v, F (v)] ± [v, F (v)]ud(v) = 0 (29) for all u, v ∈ L. Since [v, F (v)] ∈ L, thus 2u[v, F (v)] ∈ L and so 4u[v, F (v)]w ∈ L for all u, v, w ∈ L. Therefore, replacing u by 4u[v, F (v)]w, where w ∈ L, in (29) and then since characteristic of R is not 2, we obtain d(v)u[v, F (v)]w[v, F (v)] ± [v, F (v)]u[v, F (v)]wd(v) = 0 (30) for all u, v, w ∈ L. Right multiplying (29) by w[v, F (v)] and then subtracting from (30), we get [v, F (v)]u([v, F (v)]wd(v) −d(v)w[v, F (v)]) = 0 (31) for all u, v, w ∈ L. By Lemma 2.2, for each v ∈ V , either [v, F (v)] = 0 or [v, F (v)]wd(v) − d(v)w[v, F (v)] = 0 for all w ∈ L. Since the first case implies the second case, we conclude that [v, F (v)]wd(v) −d(v)w[v, F (v)] = 0 (32) for all v, w ∈ L. Now (32) and (29) together implies that 2d(v)u[v, F (v)] = 0 for all u, v ∈ L. Since characteristic of R is not 2, therefore d(v)u[v, F (v)] = 0 for all u, v ∈ L. By primeness of R, for each v ∈ L, it implies that either d(v) = 0 or [v, F (v)] = 0. Let v be an element of L such that d(v) = 0. This gives F (uv) = F (u)v for all u ∈ L. Hence, (24) gives that F ([v, u]) ± [v, u] = 0 (33) for all u ∈ L. Replacing u by 2vu in (33) and using characteristic of R is not 2, we have F (v)[v, u] ±vd([v, u]) ±v[v, u] = 0 (34) 76 b. dhara, s. ghosh, g.s. sandhu for all u ∈ L. Replacing u by 2vu in (34) and using characteristic of R is not 2, we have F (v)v[v, u] ±v2d([v, u]) ±v2[v, u] = 0 (35) for all u ∈ L. Left multiplying (34) by v and then subtracting from (35), we obtain [F (v), v][v, u] = 0 (36) for all u ∈ L. Replacing u by 2tu in (36) and then using characteristic of R is not 2 and (36), we have [F (v), v]t[v, u] = 0 for all t, u ∈ L. Since R is prime, [F (v), v] = 0 or [v, U] = (0). By Lemma 2.1, [v, U] = (0) implies v ∈ Z(R). Thus in any case, we have [F (v), v] = 0 for all v ∈ L. This implies by Lemma 2.9, L ⊆ Z(R), a contradiction. Theorem 3.4. Let L be a nonzero square closed Lie ideal of R and F be a generalized derivation of R associated to nonzero derivation d of R. If F (u◦v) ± [d(u), F (v)] ±u◦v = 0 for all u, v ∈ L, then L ⊆ Z(R). Proof. We assume on the contrary that L 6⊆ Z(R). We begin with the situation F (u◦v) ± [d(u), F (v)] ±u◦v = 0 (37) for all u, v ∈ L. Replacing u by 2uv in (37), we obtain 2(F (u◦v)v + (u◦v)d(v) ± [d(u)v, F (v)] ±[ud(v), F (v)] ± (u◦v)v) = 0 (38) for all u, v ∈ L. Since characteristic of R is not 2, we have F (u◦v)v + (u◦v)d(v) ± [d(u)v, F (v)] ±[ud(v), F (v)] ± (u◦v)v = 0 (39) that is F (u◦v)v + (u◦v)d(v) ± [d(u), F (v)]v ±d(u)[v, F (v)] ± [ud(v), F (v)] ± (u◦v)v = 0 (40) on lie ideals satisfying certain differential identities 77 for all u, v ∈ L. Right multiplying (37) by v and then subtracting from (40), we get (u◦v)d(v) ±d(u)[v, F (v)] ± [ud(v), F (v)] = 0 (41) for all u, v ∈ L. Replacing u by 2vu in (41) and then using characteristic of R is not 2, we obtain v(u◦v)d(v) ± (vd(u) + d(v)u)[v, F (v)] ±v[ud(v), F (v)] ± [v, F (v)]ud(v) = 0 (42) for all u, v ∈ L. Left multiplying (41) by v and then subtracting from (42), we get d(v)u[v, F (v)] + [v, F (v)]ud(v) = 0 (43) for all u, v ∈ L. Identity (43) coincides with identity (29) so arguing as in Theorem 3.3 we get d(v)u[v, F (v)] = 0 for all u, v ∈ L. By primeness of R, for each v ∈ L, it implies that either d(v) = 0 or [v, F (v)] = 0. Let v be an element of L such that d(v) = 0. This gives F (uv) = F (u)v for all u ∈ L. Hence, (37) gives that F (v ◦u) ±v ◦u = 0 (44) for all u ∈ L. Replacing u by 2vu in (44) and using characteristic of R is not 2, we have F (v)(v ◦u) ±vd(v ◦u) ±v(v ◦u) = 0 (45) for all u ∈ L. Replacing u by 2vu in (45) and using characteristic of R is not 2, we have F (v)v(v ◦u) ±v2d(v ◦u) ±v2(v ◦u) = 0 (46) for all u ∈ L. Left multiplying (45) by v and then subtracting from (46), we obtain [F (v), v](v ◦u) = 0 (47) for all u ∈ L. Replacing u by 2wu in (47) and then using characteristic of R is not 2, we have 0 = [F (v), v](v ◦wu) = [F (v), v]w[u, v] + [F (v), v](v ◦w)u 78 b. dhara, s. ghosh, g.s. sandhu for all u, w ∈ L. Since [F (v), v](v ◦ w) = 0 for all v, w ∈ L, we have [F (v), v]w[u, v] = 0 for all u, v, w ∈ L. Since R is prime, [F (v), v] = 0 or [v, L] = (0). By Lemma 2.1, [v, L] = (0) implies v ∈ Z(R). Thus in any case, we have [F (v), v] = 0 for all v ∈ L. This implies by Lemma 2.9, that L ⊆ Z(R), which is a contradiction. Theorem 3.5. Let L be a nonzero square closed Lie ideal of R and F and G be two generalized derivations of R associated to derivations d(6= 0) and g of R respectively. If F (u)G(v) ± d(u)F (v) ± uv ∈ Z(R) for all u, v ∈ L and d±g 6= 0, then L ⊆ Z(R). Proof. We assume on the contrary that L 6⊆ Z(R). By hypothesis, we have F (u)G(v) ±d(u)F (v) ±uv ∈ Z(R) (48) for all u, v ∈ L. Replacing v by 2vw in (48), we have 2(F (u)G(v) ±d(u)F (v) ±uv)w + 2F (u)vg(w) ± 2d(u)vd(w) ∈ Z(R) (49) for all u, v, w ∈ L. Since characteristic of R is not 2, we have (F (u)G(v) ±d(u)F (v) ±uv)w + F (u)vg(w) ±d(u)vd(w) ∈ Z(R) (50) for all u, v, w ∈ L. Commuting both sides of (50) with w, we obtain [(F (u)G(v) ±d(u)F (v) ±uv)w, w] +[F (u)vg(w) ±d(u)vd(w), w] = 0 (51) for all u, v, w ∈ L. Using (48), we obtain [F (u)vg(w) ±d(u)vd(w), w] = 0 (52) for all u, v, w ∈ L. Replacing u by 2uw in (52) and then using characteristic of R is not 2, we have [F (u)wvg(w), w] ± [d(u)wvd(w), w] +[ud(w)v(g(w) ±d(w)), w] = 0 (53) for all u, v, w ∈ L. Replacing v by 2wv in (52) and using characteristic of R is not 2, we have [F (u)wvg(w), w] ± [d(u)wvd(w), w] = 0 (54) on lie ideals satisfying certain differential identities 79 for all u, v, w ∈ L. Subtracting (54) from (53), we obtain [ud(w)v(g(w) ±d(w)), w] = 0 for all u, v, w ∈ L. Replacing u by 2tu and then using characteristic of R is not 2, it gives [t, w]ud(w)v(g(w) ±d(w)) = 0 (55) for all u, v, w, t ∈ L. Thus for each w ∈ L, either [L, w] = (0) or d(w) = 0 or d(w) ±g(w) = 0. Now the first case i.e., [L, w] = (0) implies w ∈ Z(R) by Lemma 2.1. Then by (50), we have F (u)vg(w) ±d(u)vd(w) ∈ Z(R) (56) for all u, v ∈ L. Replacing u by 2ut, where t ∈ L, we get 2(F (u)tvg(w) + ud(t)vg(w) ±d(u)tvd(w) ±ud(t)vd(w)) ∈ Z(R) (57) for all u, v, t ∈ L. Now we replace v by 2tv in (56) and obtain 2(F (u)tvg(w) ±d(u)tvd(w)) ∈ Z(R) (58) for all u, v, t ∈ L. Subtracting (58) from (57) and then using characteristic of R is not 2, we have ud(t)v(g(w) ±d(w)) ∈ Z(R) (59) for all u, v, t ∈ L. Since g(w) ± d(w) ∈ Z(R) for w ∈ Z(R), we have 0 = [ud(t)v(g(w)±d(w)), r] = [ud(t)v, r](g(w)±d(w)) for all u, v, t ∈ L and r ∈ R. Since center of a prime ring contains no divisor of zero, either [ud(t)v, r] = 0 for all u, v, t ∈ L and r ∈ R or g(w) + d(w) = 0. We consider the first case i.e., [ud(t)v, r] = 0 for all u, v, t ∈ L and r ∈ R. Replacing u by 2su, where s ∈ L, we obtain 0 = 2[sud(t)v, r] = 2s[ud(t)v, r] + 2[s, r]ud(t)v = 2[s, r]ud(t)v for all u, v, t, s ∈ L and r ∈ R. By Lemma 2.2, either [L, R] = (0) or d(L) = (0). By Lemma 2.3 both cases give L ⊆ Z(R), a contradiction. Thus we conclude that for each w ∈ L, either d(w) = 0 or d(w)±g(w) = 0. But the sets {w ∈ L : d(w) = 0} and {w ∈ L : d(w) ± g(w) = 0} are two additive subgroups of L whose union is L. By using the same argument used in Lemma 2.9 we get that either d(L) = (0) or (d ± g)(L) = (0). Since L is noncentral, by Lemma 2.3, either d = 0 or d±g = 0, a contradiction. 80 b. dhara, s. ghosh, g.s. sandhu Theorem 3.6. Let L be a nonzero square closed Lie ideal of R and F and G be generalized derivations of R associated to derivations d and g of R, respectively. If d 6= 0, G(uv) ± d(u)F (v) ± F (vu) = 0 for all u, v ∈ L, then L ⊆ Z(R). Proof. We assume on the contrary that L 6⊆ Z(R). By hypothesis, we have G(uv) ±d(u)F (v) ±F (vu) = 0 (60) for all u, v ∈ L. Replacing v by 2vu in the above relation and using char(R) 6= 2, we get (G(uv) ±d(u)F (v) ±F (vu))u + uvg(u) ±d(u)vd(v) ±vud(u) = 0. (61) Using (60) this gives uvg(u) ±d(u)vd(v) ±vud(u) = 0 (62) for all u, v ∈ L. Again replacing v by 2uv and using characteristic of R is not 2, we have u2vg(u) ±d(u)uvd(v) ±uvud(u) = 0 (63) for all u, v ∈ L. Left multiplying (62) by u and subtracting from (63), we have [d(u), u]vd(u) = 0 (64) for all u, v ∈ L. By primeness of R, for each u ∈ L, we have either [d(u), u] = 0 or d(u) = 0. Thus in each case, we have [d(u), u] = 0 for all u ∈ L. Now if d 6= 0, by Lemma 2.9, [d(u), u] = 0 for all u ∈ L implies L ⊆ Z(R), a contradiction. Theorem 3.7. Let L be a nonzero square closed Lie ideal of R and F be a generalized derivation of R associated to the nonzero derivation d of R. If F (uv) ±F (v)F (u) ±uv ∈ Z(R) for all u, v ∈ L, then L ⊆ Z(R). Proof. We assume on the contrary that L 6⊆ Z(R). First we consider the relation F (uv) + F (v)F (u) + uv ∈ Z(R) (65) for all u, v ∈ L. Replacing u with 2uw in (65) and using characteristic of R is not 2, we have F (u)wv + ud(wv) + F (v)(F (u)w + ud(w)) + uwv ∈ Z(R) (66) on lie ideals satisfying certain differential identities 81 for all u, v, w ∈ L. Commuting with w, we have [F (u)wv, w] + [ud(wv), w] + [F (v)F (u), w]w +[F (v)ud(w) + uwv, w] = 0 (67) for all u, v, w ∈ L. From (65), we can write that [F (uv)+F (v)F (u)+uv, w] = 0 for all u, v, w ∈ L, that is, [F (v)F (u), w] = −[F (uv)+uv, w] for all u, v, w ∈ L. Using this (67) reduces to [F (u)wv, w] + [ud(wv), w] − [F (uv) + uv, w]w +[F (v)ud(w) + uwv, w] = 0 (68) for all u, v, w ∈ L. Replacing v by w2 in (68), we have [F (u)w3, w] + [ud(w3), w] − [F (uw2) + uw2, w]w +[F (w2)ud(w) + uw3, w] = 0 (69) that is, [uw2d(w), w] + [F (w2)ud(w), w] = 0 (70) for all u, w ∈ L. Replacing u by 2wu in (68) and using characteristic of R is not 2, we have [(F (w)u + wd(u))wv, w] + [wud(wv), w] −[F (w)uv + wd(uv) + wuv, w]w + [F (v)wud(w) + wuwv, w] = 0 that is, [F (v)wud(w) + wd(u)wv + wud(wv) −wd(uv)w −wuvw + wuwv, w] +[F (w)uwv, w] − [F (w)uv, w]w = 0 for all u, v, w ∈ L. Assuming v = w, we have [F (w)wud(w) + wud(w2) + wd(u)w2 −wd(uw)w, w] = 0 (71) this gives [F (w)wud(w) + wud(w2) −wud(w)w, w] = 0 (72) for all u, w ∈ L. 82 b. dhara, s. ghosh, g.s. sandhu Subtracting (72) from (70), we get [uw2d(w), w] + [F (w2)ud(w), w] −[F (w)wud(w) + wud(w2) −wud(w)w, w] = 0 for all u, w ∈ L. This reduces to [uw2d(w), w] + [wd(w)ud(w), w] − [wuwd(w), w] = 0 (73) for all u, w ∈ L. Now Replacing u by 2wu in (73) and using characteristic of R is not 2, we get w[uw2d(w), w] + [wd(w)wud(w), w] −w[wuwd(w), w] = 0 (74) for all u, w ∈ L. Left multiplying (73) by w and then subtracting from (74), we get [w[d(w), w]ud(w), w] = 0 (75) for all u, w ∈ L. Again replacing u by 2uw in the above relation and using characteristic of R is not 2, we get [w[d(w), w]uwd(w), w] = 0 (76) for all u, w ∈ L. Now right multiplying (75) by w and then subtracting from (76), we obtain [w[d(w), w]u[d(w), w], w] = 0 (77) and hence [w[d(w), w]uw[d(w), w], w] = 0 (78) for all u, w ∈ L. This implies w[d(w), w]uw[d(w), w]w −w2[d(w), w]uw[d(w), w] = 0 (79) for all u, w ∈ L. Since L is a Lie ideal of R, [d(w), w] ∈ L for all w ∈ L and so 2[d(w), w]x ∈ L, 4w[d(w), w]x ∈ L and 8uw[d(w), w]x ∈ L for all u, w, x ∈ L. Hence, we can replace u with 8uw[d(w), w]x in (79) and then using characteristic of R is not 2 we obtain, w[d(w), w]uw[d(w), w]xw[d(w), w]w −w2[d(w), w]uw[d(w), w]xw[d(w), w] = 0 (80) on lie ideals satisfying certain differential identities 83 for all u, w, x ∈ L. Using (79), (80) gives w[d(w), w]uw2[d(w), w]xw[d(w), w] −w[d(w), w]uw[d(w), w]wxw[d(w), w] = 0 (81) that is w[d(w), w]u[w[d(w), w], w]xw[d(w), w] = 0 (82) for all u, w, x ∈ L. This implies w[[d(w), w], w]uw[[d(w), w], w]xw[[d(w), w], w] = 0 for all u, w, x ∈ L. By primeness of R, w[[d(w), w], w] = 0 for all w ∈ L. Then by Lemma 2.8, L ⊆ Z(R) if d 6= 0 which is a contradiction. The remaining identities can be proved in a similar way. Now we present an example which shows that the primeness hypothesis in the theorems is not superfluous. Example 3.1. Let Z be the ring of integers. Consider R = {( a 0 b c ) : a, b, c ∈ Z } and L = {( 0 0 b 0 ) : b ∈ Z } . Clearly, R is a ring under the usual addition and multiplication of matrices and L is a nonzero square closed Lie ideal of R. But for( 0 0 1 0 ) R ( 0 0 0 1 ) = (0), R is not prime ring. Define the maps on R as follows: F ( a 0 b c ) = ( a 0 2b 0 ) , d ( a 0 b c ) = ( 0 0 2b 0 ) ; G ( a 0 b c ) = ( a 0 b− c c ) , g ( a 0 b c ) = ( 0 0 a− c 0 ) . Then F and G are generalized derivations of R associated with nonzero derivations d and g respectively. Moreover, the conditions of Theorem 3.3, Theorem 3.4, Theorem 3.5, Theorem 3.6 and Theorem 3.7 are satisfied with F , G and d, but L 6⊆ Z(R). Hence the primeness assumption can not be removed. Acknowledgements The authors would like to thank the referee for providing very helpful comments and suggestions. 84 b. dhara, s. ghosh, g.s. sandhu References [1] S. Ali, B. Dhara, N.A. Dar, A.N. Khan, On Lie ideals with multiplica- tive (generalized)-derivations in prime and semiprime rings, Beitr. Algebra Geom., 56 (1) (2015), 325 – 337. [2] M. Ashraf, A. Ali, S. Ali, Some commutativity theorems for rings with generalized derivations, Southeast Asian Bull. Math. 31 (2007), 415 – 421. [3] M. Ashraf, A. Ali, R. Rani, On generalized derivations of prime rings, Southeast Asian Bull. Math. 29 (4) (2005), 669 – 675. [4] H.E. Bell, M.N. Daif, On commutativity and strong commutativity- preserving maps, Canad. Math. Bull. 37 (4) (1994), 443 – 447. [5] J. Bergen, I.NḢerstein, J.W. Kerr, Lie ideals and derivations of prime rings, J. Algebra 71 (1981), 259 – 267. [6] V. De Filippis, F. Rania, Commuting and centralizing generalized deriva- tions on Lie ideals in prime rings, Math. Notes 88 (5-6) (2010), 748 – 758. [7] B. Dhara, N. Rehman, A. Raza, Lie ideals and action of generalized derivations in rings, Miskolc Math. Notes 16 (2) (2015), 769 – 779. [8] B. Dhara, S. Kar, K.G. Pradhan, Generalized derivations acting as homomorphisms or anti-homomorphisms with central values in semiprime rings, Miskolc Math. Notes 16 (2) (2015), 781 – 791. [9] B. Dhara, S. Kar, S. Mondal, A result on generalized derivations on Lie ideals in prime rings, Beitr. Algebra Geom. 54 (2) (2013), 677 – 682. [10] B. Dhara, S. Ali, A. Pattanayak, Identities with generalized derivations in semiprime rings, Demonstratio Math. 46 (3) (2013), 453 – 460. [11] B. Dhara, Generalized derivations acting as a homomorphism or anti-homo- morphism in semiprime rings, Beitr. Algebra Geom. 53 (2012), 203 – 209. [12] B. Dhara, Remarks on generalized derivations in prime and semiprime rings, Int. J. Math. Math. Sci., Volume 2010, Article ID 646587, 6 pp. [13] S. Huang, Generalized derivations of prime rings, Int. J. Math. Math. Sci. Volume 2007, Article ID 85612, 6 pp. [14] P.H. Lee, T.K. Lee, Lie ideals of prime rings with derivations, Bull. Inst. Math. Acad. Sinica 11 (1983), 75 – 80. [15] M.A. Quadri, M.S. Khan, N. Rehman, Generalized derivations and commutativity of prime rings, Indian J. Pure Appl. Math. 34 (9) (2003), 1393 – 1396. [16] N. Rehman, On generalized derivations as homomorphisms and anti- homomorphisms, Glas. Mat. Ser. III 39(59) (1) (2004), 27 – 30. [17] N. Rehman, M. Hongan, Generalized Jordan derivations on Lie ideals associated with Hochschild 2-cocycles of rings, Rend. Circ. Mat. Palermo (2) 60 (2011), 437 – 444. [18] S.K. Tiwari, R.K. Sharma, B. Dhara, Identities related to general- ized derivations on ideal in prime rings, Beitr. Algebra Geom. 57 (4) (2016), 809 – 821. Introduction Preliminaries Main results