On Derivations and Commutativity in Prime Rings

Let R be a prime ring of characteristic different from 2, d a nonzero derivation of R, and I a nonzero right ideal of R such that [[d(x), x], [d(y), y]] = 0, for all x, y ∈ I. We prove that if [I, I]I ≠ 0, then d(I)I = 0. 1. Introduction. Let R be a prime ring and d a nonzero derivation of R. Define [x, y] 1 = [x, y] = xy − yx, then an Engel condition is a polynomial [x, y] k = [[x, y] k−1 ,y]

for any matrix A = i,j r ij e ij , where r i,j ∈ F .In this case, for any x, y ∈ R, but clearly R is not commutative.We will proceed by first proving the following theorem.
Theorem 1.2.Let R be a prime ring of characteristic different from 2, d a nonzero derivation of R, such that [[d(x), x], [d(y), y]] = 0, for all x, y ∈ R.Then, R is commutative.
Finally, in the second part of the paper, we will extend the previous theorem to a nonzero right ideal of R.
We will prove the following theorem.
Theorem 1.3.Let R be a prime ring of characteristic different from 2, d a nonzero derivation of R and I a nonzero right ideal of R such that [[d(x) We first fix the following facts.

Fact 1.
In what follows, we denote by Q the Martindale quotients ring of R and by C = Z(Q) the extended centroid of R (see [1,Chapter 2]).When R is prime, all that we need here about these objects is that R ⊆ Q, Q is prime, and C is a field.
Let T = Q * C C{X} be the free product over C of the C-algebra, Q, and the free C-algebra, C{X}, with X a countable set consisting of noncommuting indeterminates {x 1 ,...,x n ,... }.The elements of T are called generalized polynomial with coefficients in Q. I, IR, and IQ satisfy the same generalized polynomial identities with coefficients in Q.For more details about these objects we refer the reader to [1,3].Fact 3 (see Kharchenko [7]).Let f (x 1 ,...,x n ,d(x 1 ),...,d(x n )) be a differential identity of R. One of the following holds: (1) either d is an inner derivation in Q, in the sense that there exists q ∈ Q such that d(x) = [q, x], for all x ∈ Q and Q satisfies the generalized polynomial identity f (x 1 ,...,x n ,[q,x 1 ],...,[q,x n ]); or (2) R satisfies the generalized polynomial identity f (x 1 ,...,x n ,y 1 ,...,y n ).Fact 4 (see Lee [10]).I, IR, and IQ satisfy the same differential identities with coefficients in Q.
In all that follows, unless stated otherwise, R will be a prime ring of characteristic ≠ 2, d ≠ 0 a derivation of R and I a nonzero right ideal of R such that [[d(x), x], [d(y), y]] = 0, for all x, y ∈ I.

2.
The case I = R.In this section, we consider the case when [[d(x), x], [d(y), y]] = 0, for all x, y ∈ R and prove Theorem 1.2.
Proof of Theorem 1.2.Denote the differential polynomial (2.1) ) is a differential identity on R.
Using Fact 3, one of the following holds: (1) d is an inner derivation in Q, induced by c ∈ Q and R satisfies the generalized polynomial identity (2.2) (2) R satisfies the generalized polynomial identity g(x 1 ,x 2 ,y 1 ,y 2 ).In this last case, R satisfies the identity Since R is a polynomial identity (P.I.) ring, there exists a field F such that R and M t (F ), the ring of t × t matrices over F , satisfy the same polynomial identities.(2.3) Therefore, we must have t = 1 and so R is commutative.Now, let d be the inner derivation induced by an element c ∈ Q.Thus, for any r 1 ,r 2 ∈ R, that is, R satisfies a nontrivial generalized polynomial identity.By [11], it follows that S = RC is a primitive ring with soc(R) = H ≠ 0 and eHe is a simple central algebra finite-dimensional over C, for any minimal idempotent element e ∈ S.Moreover, we may assume H noncommutative, otherwise also R must be commutative.
Since H is a simple ring, one of the following holds: either H does not contain any nontrivial idempotent element or H is generated by its idempotents.
In this last case, suppose that H contains two minimal orthogonal idempotent elements e, f so that eH, f H are isomorphic H-modules.For all x ∈ H, (2.5) Left multiplying by e, we have −2ecf xecf xe = 0.This implies in particular (ecf x) 3 = 0. From this, by [5], ecf H = 0.By the primeness of H, this implies that, for any orthogonal idempotent elements of rank 1, e and f , ecf = 0. Hence, [c, e] = 0, for any idempotent e of rank 1, and [c, H] = 0, since H is generated by these idempotent elements.This argument gives the contradiction that c ∈ C and d = 0. Therefore, H cannot contain two minimal orthogonal idempotent elements and so H = D, for a suitable division ring D finite dimensional over its center.This implies that Q = H and c ∈ H.By [13, Theorem 2.3.29,page 131] (see also [9,Lemma 2]), there exists a field for F a field.As we have just seen, if n ≥ 2, then c ∈ C and d = 0.If n = 1, then H ⊆ F and we are also done.
On the other hand, if H does not contain any nontrivial idempotent element, then H is a finite-dimensional division algebra over C and c where F is a splitting field of H.In this case, a Vandermonde determinant argument shows that in M r (F ) is still an identity.As above, one can see that if r ≥ 2, then c commutes with any idempotent element in M r (F ).In this case, we have the contradiction d = 0.In the other one, H is commutative, as well as R.
3. The case I is a right ideal of R. In this final section, we will prove the main theorem of the paper (Theorem 1.3).
For the rest of the paper, we now assume the conclusion of Theorem 1.3 to be false; our goal is to ultimately arrive at a contradiction.Thus, we will assume henceforth that d(I)I ≠ 0. We begin with the following lemma.

Lemma 3.1. R is a ring satisfying a nontrivial generalized polynomial identity (GPI).
Proof.Suppose by contradiction that R does not satisfy any nontrivial generalized polynomial identity.We divide the proof into two cases.Expanding the previous GPI we get Suppose that {b, cb} are linearly C-dependent, then there exists 0 ≠ α ∈ C such that cb = αb.In this case, R satisfies Since b, c ∉ C, this last formula is a nontrivial generalized polynomial identity for R (see [3]), a contradiction.On the other hand, if {b, cb} are linearly C-independent it follows, again by Chuang's results in [3], that [[c, bx] 2 ,[c,by] 2 ] is a nontrivial generalized polynomial identity for R. In any case, we have a contradiction.
Case 2. Suppose now that d is an outer derivation.First, notice that if for all t ∈ I there exists α t ∈ C such that d(t) = α t t, then [d(x), x] is an identity for I.This implies the contradiction that R is commutative (as a consequence of [4]).
So Proof.By Lemma 3.1, R is GPI and so Q has nonzero socle H with nonzero right ideal J = IH [11].Note that H is simple, J = JH, and J satisfies the same basic conditions as I, in view of Fact 4. Now, just replace R by H, I by J, and we are done.Now, we are ready to prove the main result.

Proof of Theorem 1.3. Since I does not satisfy
Here, we suppose that d(I)I ≠ 0, that is, there exist a 4 ,a 5 ∈ I such that d(a 4 )a 5 ≠ 0 and we proceed to derive a contradiction.In view of Fact 3, we divide the proof into two cases.
Case 1.If d is an inner derivation induced by the element q ∈ Q, then I satisfies the identity [[q, x] 2 ,[q,y] 2 ], moreover, qI ≠ 0, since d(I)I ≠ 0. Let e 2 = e ∈ I. Thus, for all y ∈ R, [[q, e]  By Theorem 1.2, we have that either d = 0 or is commutative.Therefore, we have that either d(eR)eR = 0 or [eR, eR]eR = 0.
On the other hand, we have that [ea 1 ,ea 2 ]ea 3 = [a 1 ,a 2 ]a 3 ≠ 0 and also d(ea 4 )ea 5 = d(a 4 )a 5 ≠ 0. This contradiction completes the proof of the theorem.

Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space.Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon.Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall.This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles.His original model was then modified and considered under different approaches and using many versions.Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).
We intend to publish in this special issue papers reporting research on time-dependent billiards.The topic includes both conservative and dissipative dynamics.Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned.Mathematical papers regarding the topics above are also welcome.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable:

d e 11
x ,e 11 x , d e 11 y ,e 11 y = 0 (1.1) , x], [d(y), y]] = 0, for all x, y ∈ I.If [I, I]I ≠ 0, then d(I)I = 0.The assumption [I, I]I ≠ 0 is essential to the main result.In fact, consider Example 1.1 and notice that [x 1 ,x 2 ]x 3 is an identity for I = e 11 R, but clearly d(I)I = d(e 11 R)e 11 R ≠ 0.

Fact 2 .
Any derivation of R can be uniquely extended to a derivation of Q, and so any derivation of R can be defined on the whole of Q [1, Proposition 2.5.1].Moreover, Q is a prime ring as well as R and the extended centroid C of R coincides with the center of Q [1, Proposition 2.1.7,Remark 2.3.1].

Case 1 .
Suppose that d is an inner derivation induced by an element c ∈ Q.By the last assumption d(I)I ≠ 0, there exists an element b ∈ I such that cb ≠ 0. Thus, R satisfies the polynomial identity, [[c, bx] 2 ,[c,by] 2 ].Moreover, we may assume b ∉ C, otherwise R should satisfy [[c, x] 2 ,[c,y] 2 ] which is a nontrivial generalized polynomial identity.