EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 15, No. 2, 2022, 454-466 ISSN 1307-5543 – ejpam.com Published by New York Business Global Derivations in differentially δ-prime rings Iman Taha1,∗, Rohaidah Masri1, Ahmad Al khalaf2, Rawdah Tarmizi1 1 Department of Mathematics, Faculty of Sciences and Mathematics, Sultan Idris Education University, Tanjong Malim, Perak, Malaysia 2 Department of Mathematics and statistic, Faculty of Sciences, Imam Mohammad Ibn Saud Islamic University, Riyadh, Saudi Arabia Abstract. Let R be an associative ring with identity. In this paper we extend the J.H. Maynes results, which he treatised in [27]. In particular, we prove that if R is a δ-prime ring with charR 6= 2 and I is a nonzero δ-ideal of R, where 0 6= δ ∈ D, c ∈ R and [c, δ(c)] in the center of R, then R is commutative. 2020 Mathematics Subject Classifications: 16N60, 16W25 Key Words and Phrases: Derivation, prime ring, δ-prime ring, δ-ideal. 1. Introduction It shall be assumed throughout here, that R is an associative ring with respect to the addition (+) and the multiplication (·) with an identity, D is the set of all derivations in R. Consider Lie multiplication “[−,−]” on R, which is defined by [c, d] = cd − dc, where [c, d] is called a Lie commutator of elements c, d of R. The set [C,D] of the additive group R+ of a ring R is the Lie commutator subgroup generated by all [c, d] such that c ∈ C and d ∈ D. Obsreve that Z(R) is the center of R, C(R) is the commutator ideal generated by the set {[c, d] : c, d ∈ R}, annT = {r ∈ R : rT = 0 = Tr} the annihilator of T ⊆ R. An additive subgroup T of R is called a Lie ideal of R if [c, d] ∈ T, for all c ∈ T and d ∈ R. An aditive map δ : R → R is called a derivation on R if δ(cd) = δ(c)d+ cδ(d) for all c, d ∈ R. Furthermore, the map ∂a : R → R defined by, ∂a(c) = ac− ca, where c ∈ R is a partial derivation generated by a ∈ R i.e., ∂a(I) = [a, I] = {[a, i] : i ∈ I}. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v15i2.4344 Email addresses: tfaith80gmail.com (I. Taha), ajalkalaf@imamu.edu.sa (A. Al Khalaf), rohaidah@fsmt.upsi.edu.my (R. Masri), rawdah@fsmt.upsi.edu.my (Rawdah Tarmizi) https://www.ejpam.com 454 © 2022 EJPAM All rights reserved. I. Taha et al. / Eur. J. Pure Appl. Math, 15 (2) (2022), 454-466 455 In addition, the map δ is called centralizing on a subset I of R if [c, δ(c)] = cδ(c)− δ(c)c ∈ Z(R) ∀c ∈ I . Assume that ∆ is a non-empty subset of D. An ideal I of R, where δ(I) ⊆ I for δ ∈ ∆ is called a δ-ideal. A ring R is called δ-prime if, for any two δ-ideals I, J of R, the condition IJ = 0 implies that I = 0 or J = 0. The special case where a centralizing automorphism is a commuting automorphism is defined by cd(c) = d(c)c, c ∈ R. Likewise, d is called a semi commuting automorphism if cd(c) = d(c)c or cd(c) = −d(c)c holds for any c ∈ R. All other definitions and facts are standard and they can be found in [17], [18], [19] and [23]. Recall that, the first theorem of Posner informs us that if R is a prime ring with charR 6= 2, then a composition of two nonzero derivations is not a derivation. Many authors generalized Posner’s theorem in various ways as Bergen [6], Chebotar [11], Chuang [12] , [13], Hirano [21], Lanski [24] and Martindale [25]. Furthermore, Cree- don[15] generalized Posner’s first theorem to semiprime algebras, i.e. he showed that the composition of two nonzero derivations in any algebra S is a derivation. On the other hand, from the second Posner theorem which states: if R is a prime ring with centralizing derivation d 6= 0 on R, then R is commutative. In fact, this theorem extremely helped some researches to study the commuting deriva- tions, because every centralizing derivation is a commuting derivation. Recall that, the assumption of primeness in the second Posner theorem is necessarily, since if we take for example the ring R = S × T such that S is a commutative ring with derivation d1 and T is a non-commutative ring, then we can prove that the derivation on R given by d(s, t) = (d1(s), 0) is a non zero commuting derivation, but R is not a commutative ring. In the last fifty years, a lot of results have been obtained about commuting and cen- tralizing derivations d (d satisfies the condition [d(c), c] ∈ Z(R) for all c ∈ R). However, many authors extended it by taking a centralizing map on a ring only. In 1973, Awtar [4] studied the centralizing derivation on Lie ideals and Jordan ideals. In particular, he proved that if R is a prime ring of charR 6= 2 and T 6= {0} is a Lie ideal or Jordan ideal and subring in R, with d 6= 0 being a derivation on R, if [c, d(c)] ∈ Z(R), for all c ∈ T then R is commutative. In addition, if we assume that either T is a Lie (Jordan) ideal or a subring , then R is not necessarily commutative. That can be shown as follows: let R be a prime ring with charR 6= 2 and d 6= 0 is a derivation of R. If T is a Lie or Jordan ideal and a subring of R and if [c, d(c)] ∈ Z(R), for all c ∈ T , then the ring R is commutative. Mayne [26] got the same result, i.e. if R is a prime ring and d 6= 0 is a centralizing automorphism, then R is an integral domain. Furthermore, Mayne [27] generalized the previous results for a derivation d or an automorphism, Moreover, Mayne [28] showed that if there exists a centralizing derivation d 6= {0} or a centralizing automorphis on an ideal T 6= 0 of a prime ring R, then R is commutative. I. Taha et al. / Eur. J. Pure Appl. Math, 15 (2) (2022), 454-466 456 Also, Awtar [4] extended the derivation case on a prime ring with any characteristic. Whereas, McCrimmon [29] proved that the automorphism in Mayne’s theorem did not generalize for a semiprime ring. Likewise, Vukman [32] has extended Posner’s second theorem by proving that if d 6= 0 is a derivation on prime ring with charR 6= 2 and [[d(c), c], c] = 0, for all c ∈ R, then either d = 0 or R is commutative. In fact, this theorem has merely showed that d is commuting. In addition, in 1992, Vukman extended the second Posner’s result for an automorphism or a centralizing derivation on a Lie ideal T 6= {0}. Whereas, in 1993 Bresar [8] showed that an additive map is not centralizing on determined subsets of prime and non-commutative ring. Futhermore, Some generalizations of these results for a prime ring are contained in [20–22]. As for a semiprime ring we refer the reader to [10], [7], [9], [32] and [33]. In our current research we shall generalize the theorem of Mayne [27] and theorem of Hirano and Tominaga [21], so this generalization of the two theorems give us a new wider class of δ-prime rings and we prove the following Theorem 1. Let R be a δ-prime ring of charastristic 6= 2 and I be a nonzero δ-ideal of R, where 0 6= δ ∈ D. If [c, δ(c)] ∈ Z(R) ∀c ∈ I. Then R is commutative. 2. Preliminaries Many authors have been studying the centralizing automorphisms and derivation on ring R. C. R. Miers [30] has considered the map defined on C∗ Algebra. Moreover, in [4] R.A. Awtar showed if existence a nonzero centerlizing derivation on a prime ring, then R is commutative, so he gives a shorter proof of Posner’s theorem [31]. Awtar in [4] proved that if R is a prime ring with charR 6= 2 havig a derivation d on a Jordan ideal J 6= {0}, where the derivation is centralizing on J , implies J ⊆ Z(R). In [14] L.O. Chung and J.Luh showed the equivalence between semi-commuting au- tomorphism and commuting automorphism on a prime ring. If the prime ring R has a nontrivial semicommuting automorphism and R with charR 6= 2 or Z(R) 6= {0}, this implies the commutativity of the ring R. In [16] N. Divinsky proved that if the simple Artinian ring has a nontrivial centralizing automorphism, then R is a field. On the other hand in [21] it has been proved that if R has a nontrivial automorphism, then R is a field. In [1, 2] it has been proved the commutativity of a prime and semiprime rings. Now willing to prove our theorem, we will need to state some lemmas: Lemma 1. Let I 6= {0} be δ-ideal of a δ-prime ring R. If δ(I) = 0, then δ(R) = 0. I. Taha et al. / Eur. J. Pure Appl. Math, 15 (2) (2022), 454-466 457 Proof. Since RI ⊆ I and IR ⊆ I. Then δ(RI) = δ(R)I = 0 = δ(IR) = Iδ(R). Thus we deduce that δ(R) ⊆ annI, but I is a δ-ideal and so δ(R) = 0. Lemma 2. Let δ 6= 0 be a derivation on a ring R and I 6= {0} be δ-ideal of R. If R is δ-prime such that [δ(a), a] = 0 ∀a ∈ I. (2− 1) then R is commutative. Proof. Linearizing the equation (2-1) on I, then we have for all a, b, c ∈ I 0 = [δ(a+ b), a+ b] = [δ(a), a] + [δ(a), b]+ [δ(b), a] + [δ(b), b] = [δ(a), b] + [δ(b), a]. Thus we conclude that [δ(a), b] = [a, δ(b)]. (2− 2) Now replacing the right side in (2-2) δ(b) by aδ(b) we have [a, aδ(b)] = a[a, δ(b)] = a[δ(a), b] = aδ(a)b− abδ(a) = δ(a)ab− abδ(a) = [δ(a), ab] = [a, δ(ab)] = [a, δ(a)b] + [a, aδ(b)]. Hence δ(a)[a, b] = [a, δ(a)b] = 0. (2− 3) Now replacing b by cb in (2-3) we obtain that 0 = δ(a)[a, cb] = δ(a)[a, c]b+ δ(a)c[a, b] = δ(a)c[a, b]. Consequently, δ(a)I[a, b] = 0. Thus by using the δ-primenes we get either a = 0 or [a, b] = 0. Since I 6= {0}, then we deduce that [a, b] = 0 and therefore, I is commutative. Then we have I2C(R) = 0, and so C(R) = 0. Hence R is commutative. I. Taha et al. / Eur. J. Pure Appl. Math, 15 (2) (2022), 454-466 458 3. δ-derivation on δ-ideal First of all in the next lemma we give a generalization of Lemma 1 from [31] Lemma 3. Let δ 6= 0 be a derivation of a ring R . If R is δ-prime ring such that a[δn(a), R] = 0, (respectively [δn(b), R]a = 0) ∀a, b ∈ R, and for all integers n ≥ 0, then either a = 0 or b ∈ Z(R). Proof. Suppose that x, y ∈ R and n, k are a nonnegative integers. From [31] we have a∂δn(b)(R) = 0, then 0 = a∂δn(b)(xy) = a∂δn(b)(x)y + ax∂δn(b)(y) = ax∂δn(b)(y). This means that aR[δn(b), y] = 0. Consequently, aRδk([δn(b), y]) = 0, what forces that a = 0 or [δn(b), y] = 0 (and then b ∈ Z(R)). Lemma 4. Let I 6= {0} be a right δ-ideal of a δ-prime ring R. If I is commutative, then R is commutative. Proof. Suppose that a ∈ I. Then ∂a(I) = 0 and so ∂a(R) ⊆ annI. Since annI is a δ-ideal, then we see that annI = 0, and so a ∈ Z(R). Hence I ⊆ Z(R). Then IC(R) = 0 and hence C(R) = 0. I. Taha et al. / Eur. J. Pure Appl. Math, 15 (2) (2022), 454-466 459 Lemma 5. Let δ 6= 0 be a nonzero derivation of a δ-prime ring R. If [b, δn(a)b] ∈ Z(R), and 0 6= b ∈ R for all integers n ≥ 0, then a ∈ Z(R). Proof. Since for all x ∈ R we have 0 = [δn(a)b, x] = δn(a)[b, x] + [δn(a), x]b = [δn(a), x]b. Then by Lemma 3 we see that a ∈ Z(R). Lemma 6. Let R be a δ−prime ring of characterstic 6= 2 and I be a δ−ideal of R. If [x, δ(x)] ∈ Z(R) ∀x ∈ I, (3− 1) then [x, δ(x)] = 0. Proof. Suppose that x, y ∈ I. Now replace x by x+ y in (3-1) we get [x+ y, δ(x+ y)] = [x, δ(x)] + [x, δ(y)] + [y, δ(x)] + [y, δ(y)], and so [x, δ(y)] + [y, δ(x)] ∈ Z(R). (3− 2). Now substituting y by x2 in (3-2), we obtain 4x[x, δ(x)] = [x, δ(x2)] + [x2, δ(x)] ∈ Z(R). Consequently, x[x, δ(x)] ∈ Z(R) (3− 3). Then 0 = [x[x, δ(x)], δ(x)] = [x, δ(x)]2. Obviously that δ([x, δ(x)]) ∈ Z(R). and δ([x, δ(x)]) = [δ(x), δ(x)] + [x, δ2(x)] = I. Taha et al. / Eur. J. Pure Appl. Math, 15 (2) (2022), 454-466 460 [x, δ2(x)] and δ([x, δ2(x)]) ∈ Z(R). δ([x, δ2(x)]) = [δ(x), δ2(x)] + [x, δ3(x)]. Hence [δ(x), δ2(x)] ∈ Z(G) we have [x, δ3(x)] ∈ Z(R). In addition, by induction on n we obtain that [x, δn(x)] ∈ Z(R), (3− 4). Now substituting y by xδn(x) in (3-2) we get (since [x, δ(xδn(x)] + [xδn(x), δ(x)] ∈ Z(G)) [x, δ(xδn(x)] + [xδn(x), δ(x)] = [x, δ(x)δn(x)] + [x, xδn+1(x)]− [δ(x), xδn(x)] = [x, δ(x)]δn(x) + δ(x)[x, xδn(x)]+ +x[x, xδn+1(x)]− [δ(x), x]δn(x)− −x[δ(x), δn(x)] =: T. Then, 0 = [T, δn(x)] = [δ(x), δn(x)].[x, δn(x)]+ [x, δn(x)].[x, δn+1(x)]− [x, δn(x)].[δ(x), δn(x)] = [x, δn(x)][x, δn+1(x)]. (3− 5) Substituting y by x2δn(x) in (3-2), we will obtain [x, δ(x2δn(x))] + [x2δn(x), δ(x)] ∈ Z(R)) [x, δ(x2δn(x))] + [x2δn(x), δ(x)] = I. Taha et al. / Eur. J. Pure Appl. Math, 15 (2) (2022), 454-466 461 [x, δ(x)xδn(x)] + [x, xδ(x)δn(x)]+ +[x, x2δn+1(x)]− [δ(x), x2δn(x)] = [x, δ(x)x]δn(x) + [x, δn(x)]δ(x)x+ [x, xδ(x)]δn(x) + [x, δn(x)]xδ(x)+ [x, x2]δn+1(x) + [x, δn+1(x)]x2− −[δ(x), x2]δn(x)− [δ(x), δn(x)]x2 = [x, δ(x)]xδn(x) + [x, x]δ(x)δn(x)+ [x, δn(x)]δ(x)x+ [x, x]δ(x)δn(x)+ [x, δ(x)]xδn(x) + [x, δn(x)]xδ(x)+ [x, δn+1(x)]x2 − 2[δ(x), x]xδn(x)− −[δ(x), δn(x)]x2 = 4[x, δ(x)]xδn(x) + [x, δn(x)]xδ(x)+ +[x, δn(x)]xδ(x)x+ [x, δn+1(x)]x2− +[δ(x), δn(x)]x2 =: Q. (3− 6) multiplying Q by [x, δn(x)] in (3-6) A in view of (3-2) we obtain [x, δn(x)]2xδ(x) + [x, δn(x)]2δ(x)x− −[δ(x), δn(x)][x, δn(x)]x2 ∈ Z(R). Then, I. Taha et al. / Eur. J. Pure Appl. Math, 15 (2) (2022), 454-466 462 0 = [δn(x), F ] = [δn(x), xδ(x)][x, δn(x)]2+ +[δn(x), δ(x)x][xδn(x)]2− −[δ(x), δn(x)][x, δn(x)][δn(x), x2] = = [δn(x), x][x, δn(x)]2δ(x)+ = [δn(x), δ(x)][x, δn(x)]2δ(x)+ +[δn(x), δ(x)][x, δn(x)]2x+ +[δn(x), δ(x)][x, δn(x)]2x+ +[δn(x), x][x, δn(x)]2δ(x)− −2[δ(x), δn(x)][x, δn(x)][δn(x), x]x = 2[δn(x), δ(x)][x, δn(x)]2− −2[x, δn(x)]3δ(x) + 2[δ(x), δn(x)][x, δn(x)]2x = = 2[x, δn(x)]3δ(x)] =: X Then [x, δn(x)]3δ(x)] = 0 Thus [x, δn+1(x)]3δ(x)] = 0 Hence [x, δn+1(x)]3[δ(x), δn(x)] = [x, δn+1(x)]3δ(x)δn(x)− I. Taha et al. / Eur. J. Pure Appl. Math, 15 (2) (2022), 454-466 463 −δn(x)[x, δn+1(x)]3δ(x) = 0. (3− 7) Multiplying (3-7) by [x, δn+1(x)]2 we obtain [x, δn+1(x)]4 = 0. Thus ([x, δn+1(x)]R4 = 0. this means that A = ∞∑ n=1 ∑ x∈I [x, δn(x)]R. (δ([x, δn+1(x)]R) = [δ(x), δn+1(x)]R+ [x, δn+2(x)]R+ [x, δn(x)]δ(R) ⊆ A) is a sum of nilpotent ideals and I is a nil ideal, since A is δ-ideal, we deduce that A = 0. This gives that [x, δ(x)] = 0. Lemma 7. Let R be δ-prime ring of charR 6= 2 and [δ(x), x] ∈ Z(R) ∀x ∈ R. Then R is commutative. Proof. It is well known that [R,R] is a Lie ideal of R. Moreover, δ([R,R]) ⊆ [R,R]. Now on the one hand if [R,R] is commutative, then by Lemma (1-7) in [5] C(R) is a nil ideal, Thus C(R) = 0, and R is commutative. On the other hand by Lemma 13 [3] [R,R] contains a nonzro δ-ideal I of R. Thus by(3-2) we have [δ(x), y] ∈ Z(R) ∀x, y ∈ R. REFERENCES 464 This means that δ(I) ⊆ Z(R).Then for all a ∈ I [δ(a), a] = 0, and by Lemma 2 R is commutative. proof of Theorem (1) Since [x, δ(x)] ∈ Z(R) for all x ∈ I, then by Lemma 7 we get [x, δ(x)] = 0. Now using Lemma 2 and since R is A δ-prime ring and [x, δ(x)] = 0, then R is commutative. Acknowledgements This project was funded by National Plan for Science, Technology and Innovation (MAARIFAH) — King Abdul Aziz City for Science and Technology — The Kingdom of Saudi Arabia, award number (14-MAT273-08 R). References [1] ��A Alkhalaf, O Artemovych, and I Taha. 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