Notes on the Higher Derivations of Prime Rings

The main purpose of these notes investigate some certain properties and relation between the higher derivation (HD, for short) and Lie ideal of semiprime ring and prime rings, we gave some results about that.


Introduction
For almost 50 years, the study of Lie isomorphisms and Lie derivations was carried on mainly by W.S.Martindal III and his students.In (1964) Martindale generalized an unpublished results of Kaplansky(obtained in the case of matrix ring over field), described Lie derivations of primitive ring of characteristic not 2 with nontrivial idempotents [1].In subsequent papers of several authors, the analogues problem was considered either in the context of prime rings with involution [2] or in the context of von Neumann algebras under a similar assumption.In the year (1961) Herstein [3] in his AMS hour talk about Lie and Jordan Structure in Simple, Associative Rings, posted a number of problems on Lie (Jordan) isomorphisms and derivations.Beidar and Chebotar [4] consider the Lie derivations of prime rings.Banning and Mathieu [5] extended to semiprime rings the description of Lie derivations obtained by Bresar in prime case.Villena [6]considered D a Lie derivation on an unital complex Banach algebra then for every primitive ideal P of A, except for a finite of them which have finite codimension greater than one ,there exists a derivation d from A/P to itself and a linear functional τ on A such that Q q D(a) = d(a+p)+τ (a) for all a ∈ A(where Q q denotes the quotient map from A onto A/P ).The relation between usual derivations and Lie ideal of prime rings has been extensively studied in the last 30 years.In particular ,when this relationship involves the action of the derivations on Lie ideals.Many of these results extend other ones proven previously just for the action of the derivations on the whole ring.In (1984), Awtar [7] extended to Lie ideal a well-know result proved by Herstein concerning derivations in prime rings.In fact, Awtar proved that if U is a Lie ideal of prime ring R of characteristic different of 2 such that u 2 ∈ U for every u ∈ U and d : R −→ R is an additive mapping such that d\U is a Jordan derivation of U into R, then d\U is a derivation of U into R, where the additive mapping d : R −→ R is called Jordan derivation when d satisfying the condition that d(x 2 ) = d(x)x + xd(x) for all x ∈ R. Lanski and Montgomery [8] [9] proved, let R be a prime ring of characteristic different from 2, L a noncentral Lie ideal of R, H and G two nonzero generalized derivations of R. Suppose us(H(u)u − uG(u))ut = 0 for all u ∈ L, where s, t ≥ 0 are fixed integers.Then either (i) there exists p ∈ U such that H(x) = xp for all x ∈ R and G(x) = px for all x ∈ R unless R satisfies S 4 , the standard identity in four variables; or (ii) R satisfies S 4 and there exist p, q ∈ U such that H(x) = px + xq for all x ∈ R and G(x) = qx + xp for all x ∈ R. Öznur and Emine [10] proved, let R be a prime ring with characteristic different from two,U a nonzero Lie ideal of R and f be a generalized derivation associated with d.They prove the following results: for all u, v ∈ U , then U ⊂ Z(R).Vincenzo De Filippis, Nadeem UR Rehman, and Abu Zaid Ansari [11] proved, let be a 2-torsion free ring and let L be a noncentral Lie ideal of R and let F : R −→ R and G : R −→ R be two generalized derivations of R, they analyze the structure of R in the cases:R is prime and R is semiprime ring after satisfy some conditions.In this notes we gave some results about the higher derivation (HD, for short) and Lie ideal of semiprime ring and prime rings.

Preliminaries
Throughout, let R be an associative ring with the center Z(R).For any a, b ∈ R, the symbol [a, b] stands for the commutater ab − ba.Given two subset A, B of R,[A, B] will denote the additive subgroups of R generated by all elements of the form [a, b],where a ∈ A, b ∈ B. During these notes we suppose that R is a prime ring i.e. if aRb = 0 this implies to either a = 0 or b = 0 and semiprime if xRx = 0 implies x = 0.In fact, a prime ring is semiprime but the converse is not true in general.A ring R is n-torsion free, where n > 1 is an integer in case nx = 0 implies that x = 0 for any Every derivation is Jordan derivation.The converse is in general not true.A derivation d is inner in case there exists a ∈ R, such that d(x) = [a, x] holds for all x ∈ R.

The main results
Theorem 3.1.Let R be a prime ring,U be a Lie ideal of R and D = ( Thus depend on Lemma 1, we complete the proof.✷ Depend on Theorem 3.1 and Lemma 2, we can easy prove the following corollary.
Corollary 3.2.Let R be a prime ring, U be a Lie ideal of R and D = ( Let R be a prime ring, U be a Lie ideal of R and D = ( Proof.According to Theorem 3.1, we obtain d i (U ) ⊆ Z(R), so for all x, y ∈ U , we get Proof.By use the same method in Theorem 3.2 with depend on the fact, every central ideal is commutative, we complete the proof.✷ Theorem 3.5.Let R be a 2-torsion free prime ring, U be a Lie ideal of R and such that, Notes on the Higher Derivations of Prime Rings

65
According to our assumption that d i d j (U ) ⊆ Z(R), i, j ∈ N , and with R be a 2-torsion free prime ring, we obtain [d n (U ), d n (U )] ⊆ Z(R).Thus, we complete the by same technique of Theorem 3.2.✷ Theorem 3.6.Let R be a prime ring, U be a Lie ideal of R and D = ( , where a ∈ R. Proof.Suppose that a ∈ Z(R), then we have either a = 0 or d(U ) ⊆ Z(R), which implies that U ⊆ Z(R).We, now suppose that a not in Z(R),with proceed to show that U ⊆ Z(R).According to the hypothesis , which leads to the relation Then, we have ξ − η = 0. Now, let and Furthermore, we suppose that Substituting d α (abc) by Notes on the Higher Derivations of Prime Rings 63 Moreover, if U be a Lie ideal of R, D is said to be higher a derivation(HD, for short) of U into R if for every n ∈ N , we have d n (xy) = i+j=n d i (x)d j (y) for allx, y ∈ U , Jordan higher derivation (JHD, for short) of U into R if for every n ∈ N , we haved n (x 2 ) = i+j=n d i (x)d j (x)for all x ∈ U , where N denotes the set of natural numbers including 0, and D = (d i = 0) for all i ∈ N is the family of additive mappings of R such that d 0 = id R .We denote by τ n (a, b, c) the element of R is defined by τ n (a, b, c) = d n (a, b, c) − i+j+k=n d i (a)d j (b)d k (c).Then τ n is an additive in each argument.To achieve our purpose, we mention the following lemmas.Lemma 2.1.[12] If U is a Lie ideal of a semiprime ring R and [U, U ] ⊆ Z(R),then U ⊆ Z(R).

Lemma 2 . 2 .
[13, Lemma3]  If a prime ring R contains a non-zero commutative right ideal, then R is commutative.Lemma 2.3.[14]Let R be a 2-torsion free semiprime ring (resp.prime ring and U an admissible Lie ideal of R).Then τ n (a, b, c) = 0 for every a, b, c ∈ R(resp.a,b, c ∈ U ),n ∈ N, N denotes the set of natural numbers.

Corollary 3 . 4 .
the above relation reduces to d n (x)[x, y] = 0 for all x, y ∈ U .Left multiplying by r, we obtain rd n (x)[x, y] = 0 for all x, y ∈ U, r ∈ R. Since d n (x) ∈ Z(R), therefore, we get d n (x)r[x, y] = 0 for all x, y ∈ U, r ∈ R. Then from above equation, we achieved n (x)R[x, y] = 0. Since R is prime ring, then either d n (x) = 0 or [x, y] = 0.The additive group of U is the union of two subgroups, then Kerd n (kernel of d n ) and Z(U ) is the center of U , so one of them must be whole group.According to our hypothesis d i = 0 for all i ∈ N, Kerd n = U .Thus Z(U ) = U , therefore, U is commutative.According to Lemma 2, we complete the proof.✷Let R be a prime ring, U be a Lie ideal of R and D = (

Corollary 3 . 7 .
Again by using that ad i (U ) ⊆ Z(R) and ϑ ∈ Z(R), with d i (ϑ) = 0 ∈ Z(R),wearrive d n (ϑ)a([x, y]) ∈ Z(R), then we have a[x, y] ∈ Z(R) for all x ∈ U, y ∈ R, which implies that a[U, R] ⊆ Z(R).Depend on the style of the above the proof of following corollary is evident.✷Let R be a prime ring, U be a Lie ideal of R and D = (d i = 0) i∈N a HD of U into R such that ad i (U ) ⊆ Z(R)and d i (Z(R)) = 0 then either a = 0 or a[U, R] ⊆ Z(R), where a ∈ R. Lemma 3.8.Let R be a 2-torsion free ring and U is a Lie ideal of R, D = (d i ) i∈N a JHD of R into R and n ∈ N , then d n (abc + cda) = h+j+k=n d h (a)d j (b)d k (c) + d h (c)d j (b)d k (a) for all a, b, c ∈ R. Proof.First of all, we denote by ξ n (a, b, c) to the element of R is defined by ξ n (a, b, c) = d n (abc) − h+j+k=n d h (a)d j (b)d k (c).Then ξ n is additive in each argument and ξ n (a, b, c) = 0 for all a, b, c ∈ R. So, we have i+f +h=α d i (a)d j (b)d h (c), where α < n.Similarly for d α (cba), where ψ < n.Then η 1 − ξ 1 = τ n (a, b, c)rcba + abcrτ n (c, b, a), and η 1 − ξ 2 = τ n (c, b, a)rabc + cbarτ n (a, b, c).Thus, according to our hypothesis which is η − ξ = 0, we get τ n (a, b, c)rcba + abcrτ n (c, b, a) + τ n (c, b, a)rabc + cbarτ n (a, b, c) = 0. Apply the Corollary 3.5, we obtain τ n (a, b, c)r[a, b, c]+[a, b, c]rτ n (a, b, c) = 0. Thus we complete the proof of our theorem.✷ Depend on same technique in above theorem with use Lemma 3, we can prove the following results.Theorem 3.11.Let R be a 2-torsion free semiprime ring and U is a Lie ideal of R, D = (d i ) i∈N a JHD of R into R and n ∈ N , then τ n (a, b, c)[a, b, c] + [a, b, c]rτ n (a, b, c) = 0 for every a, b, c, r ∈ R, where there exists m < n, m ∈ N .Corollary 3.12.Let R be a 2-torsion free prime ring and U is a dmissible Lie ideal of R, D = (d i ) i∈N a JHD of R into R and n ∈ N , then τ n (a, b, c)[a, b, c] + [a, b, c]rτ n (a, b, c) = 0 for every a, b, c, r ∈ U , where there exists m < n, m ∈ N .