Abstract
Let be a prime ring of char , a nonzero derivation of and a nonzero right ideal of such that for all , where , , are fixed integers. If , then .
1. Introduction
Throughout this paper, unless specifically stated, always denotes a prime ring with center and extended centroid , the Martindale quotients ring. Let be a positive integer. For given , let and let be the usual commutator and inductively for , . By we mean a nonzero derivation in .
A well-known result proven by Posner [1] states that if for all , then is commutative. In [2], Lanski generalized this result of Posner to the Lie ideal. Lanski proved that if is a noncommutative Lie ideal of such that for all , then either is commutative or char and satisfies , the standard identity in four variables. Bell and Martindale III [3] studied this identity for a semiprime ring . They proved that if is a semiprime ring and for all in a non-zero left ideal of and , then contains a non-zero central ideal. Clearly, this result says that if is a prime ring, then must be commutative.
Several authors have studied this kind of Engel type identities with derivation in different ways. In [4], Herstein proved that if char and for all , then is commutative. In [5], Filippis showed that if is of characteristic different from and a non-zero right ideal of such that and for all , then .
In continuation of these previous results, it is natural to consider the situation when for all , are fixed integers. We have studied this identity in the present paper.
It is well known that any derivation of a prime ring can be uniquely extended to a derivation of , and so any derivation of can be defined on the whole of . Moreover is a prime ring as well as and the extended centroid of coincides with the center of . We refer to [6, 7] for more details.
Denote by the free product of the -algebra and , the free -algebra in noncommuting indeterminates .
2. The Case: Prime Ring
We need the following lemma.
Lemma 2.1. Let be a non-zero right ideal of and a derivation of . Then the following conditions are equivalent: (i) is an inner derivation induced by some such that ; (ii) (for its proof refer to [8, Lemma]).
We mention an important result which will be used quite frequently as follows.
Theorem 2.2 (see Kharchenko [9]). Let be a prime ring, a derivation on and a non-zero ideal of . If satisfies the differential identity then either (i) satisfies the generalized polynomial identity or (ii) is -inner, that is, for some and satisfies the generalized polynomial identity
Theorem 2.3. Let be a prime ring of char and a derivation of such that for all , where are fixed integers. Then is commutative or .
Proof. Let be noncommutative. If is not -inner, then by Kharchenko's Theorem [9]
for all . This is a polynomial identity and hence there exists a field such that with and and satisfy the same polynomial identity [10,Lemma ]. But by choosing and , we get
which is a contradiction.
Now, let be -inner derivation, say for some , that is, for all , then we have
for all . Since , and hence satisfies a nontrivial generalized polynomial identity (GPI). By [11], it follows that is a primitive ring with and is finite dimensional over for any minimal idempotent . Moreover we may assume that is noncommutative; otherwise, must be commutative which is a contradiction.
Notice that satisfies (see [10, Proof of Theorem ]). For any idempotent and we have
Right multiplying by , we get
This implies that . Since char , . By Levitzki's lemma [12, Lemma ], for all . Since is prime ring, , that is, for any idempotent . Now replacing with , we get that that is, . Therefore for any idempotent , we have . So commutes with all idempotents in . Since is a simple ring, either is generated by its idempotents or does not contain any nontrivial idempotents. The first case gives contradicting . In the last case, is a finite dimensional division algebra over . This implies that and . By [10,Lemma ], there exists a field such that and satisfies . Then by the same argument as earlier, commutes with all idempotents in , again giving the contradiction , that is, . This completes the proof of the theorem.
Theorem 2.4. Let be a prime ring of char , a non-zero derivation of and a non-zero right ideal of such that for all , where are fixed integers. If , then .
We begin the proof by proving the following lemma.
Lemma 2.5. If and for all are fixed integers, then satisfies nontrivial generalized polynomial identity (GPI).
Proof. Suppose on the contrary that does not satisfy any nontrivial GPI. We may assume that is noncommutative; otherwise, satisfies trivially a nontrivial GPI. We consider two cases.Case 1. Suppose that is -inner derivation induced by an element . Then for any
is a GPI for , so it is the zero element in . Expanding this, we get
where . If and are linearly -independent for some then
Again, since and are linearly -independent, above relation implies that
and so
Repeating the same process yields
in . This implies that , a contradiction. Thus for any , and are -dependent. Then for some . Replacing with , we may assume that . Then by Lemma 2.1, , contradiction.
Case 2. Suppose that is not -inner derivation. If for all , , then which implies that is commutative (see [13]). Therefore there exists such that , that is, and are linearly -independent.
By our assumption, we have that satisfies
By Kharchenko's Theorem [9],
for all . In particular for ,
which is a nontrivial GPI for , because and are linearly -independent, a contradiction.
We are now ready to prove our main theorem.
Proof of Theorem 2.4. Suppose that then we derive a contradiction. By Lemma 2.5, is a prime GPI ring, so is also by [14]. Since is centrally closed over , it follows from [11] that is a primitive ring with .
By our assumption and by [7], we may assume that
is satisfied by and hence by . Let and . Then replacing with and with in (2.17), then right multiplying it by we obtain that
Now we have the fact that for any idempotent , , and so
Now since for any idempotent and for any , , above relation gives
This implies that for all . Since char , we have by Levitzki's lemma [12,Lemma ] that for all . By primeness of , . By [15,Lemma ], since is a regular ring, for each , there exists an idempotent such that and . Hence gives and so and . Hence for each , . Thus . Set . Then a prime -algebra with the derivation such that , for all . By assumption, we have that
for all . By Theorem 2.3, we have either or is commutative. Therefore we have that either or . Now implies that and so . implies that and so then implying that . Thus in all the cases we have contradiction. This completes the proof of the theorem.
3. The Case: Semiprime Ring
In this section we extend Theorem 2.3 to the semiprime case. Let be a semiprime ring and be its right Utumi quotient ring. It is well known that any derivation of a semiprime ring can be uniquely extended to a derivation of its right Utumi quotient ring and so any derivation of can be defined on the whole of [7,Lemma ].
By the standard theory of orthogonal completions for semiprime rings, we have the following lemma.
Lemma 3.1 (see [16, Lemma and Theorem ] or [7,pages 31-32]). Let be a -torsion free semiprime ring and a maximal ideal of . Then is a prime ideal of invariant under all derivations of . Moreover, .
Theorem 3.2. Let be a -torsion free semiprime ring and a non-zero derivation of such that for all , fixed are integers. Then maps into its center.
Proof. Since any derivation can be uniquely extended to a derivation in and and satisfy the same differential identities [7, Theorem ], we have for all . Let be any maximal ideal of such that is -torsion free. Then by Lemma 3.1, is a prime ideal of invariant under . Set Then derivation canonically induces a derivation on defined by for all . Therefore, for all . By Theorem 2.3, either or , that is, or . In any case for any maximal ideal of . By Lemma 3.1, . Thus . Without loss of generality, we have . This implies that Therefore . By semiprimeness of , we have , that is, . This completes the proof of the theorem.