Table of Contents
International Scholarly Research Notices
Volume 2014 (2014), Article ID 563284, 9 pages
http://dx.doi.org/10.1155/2014/563284
Research Article

Two-Sided Annihilator Condition with Generalized Derivations on Multilinear Polynomials

Department of Mathematics, Faculty of Sciences, University of Messina, Via F. Stagno D’Alcontres 31, 98166 Messina, Italy

Received 30 April 2014; Accepted 15 July 2014; Published 28 October 2014

Academic Editor: Ali Jaballah

Copyright © 2014 V. De Filippis et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let be a prime ring of characteristic different from 2, with extended centroid , its two-sided Utumi quotient ring, a nonzero generalized derivation of , a noncentral multilinear polynomial over in noncommuting variables, and such that for any . Then one of the following holds: (1) ; (2) ; (3) there exists such that , for all ; (4) there exist and such that , for all , and is central valued on ; (5) there exist and such that , for all , and , .

1. Introduction

Let be a prime ring with center . We denote by the simple commutator of the elements and by , for , the th commutator of . Throught this paper we will use the following notation: will be the (two-sided) Utumi quotient ring of a ring (sometimes, as in [1], is called the symmetric ring of quotients). The definition, the axiomatic formulation, and the properties of this quotient ring can be found in [13].

In any case, when is a prime ring, all that we need here about this object is that(1);(2) is a prime ring;(3)the center of , denoted by is a field which is called the extended centroid of .A well known result of Posner [4] says that if is a derivation of such that , for all , then is commutative. In [5] Lanski generalizes the result of Posner, by replacing the element with an element of a noncentral Lie ideal of . More precisely he proves that if for all and a fixed integer; then and satisfies , the standard identity of degree .

Let be a multilinear polynomial over in noncommuting variables and denote by the set of all evaluations of in . In case is not central valued on , it is well known that the additive subgroup generated by contains a noncentral Lie ideal of . Moreover any noncentral Lie ideal of contains all the commutators for in some nonzero ideal of , unless and .

In light of this and following the line of investigation of the previous cited papers, in [6] P. H. Lee and T. K. Lee consider the Engel-condition , in case , where is a two-sided ideal of . They show that either is central valued in or and satisfies .

These results indicate that the global structure of a prime ring is often tightly connected to the behaviour of additive mappings defined on , which act on suitable subsets of the whole ring. In [7] de Filippis and di Vincenzo study the left annihilator of the set , where is a derivation. In case the annihilator is not zero, the conclusion is that is central valued on . These facts in a prime ring are natural tests which evidence that the set is rather large in .

More recently, Liu [8] and Wang [9] have examined the identity , where is a derivation of and , where is a one-sided ideal of . In particular, for , if and is not central valued on , then and satisfies .

In [10] de Filippis considers a similar situation, in the case the derivation is replaced by a generalized derivation . An additive map is said to be a generalized derivation if there is a derivation of such that, for all , . A significative example is a map of the form , for some ; such generalized derivations are called inner. Generalized derivations have been primarily studied on operator algebras. Therefore any investigation from the algebraic point of view might be interesting (see, e.g., [11]).

The main result in [10] is the following.

Theorem A. Let be a prime ring of characteristic different from 2, with extended centroid , its two-sided Utumi quotient ring, a nonzero generalized derivation of , a noncentral multilinear polynomial over in n noncommuting variables, and such that for any . Then either or one of the following holds: (1)there exists such that , for all ;(2)there exist and such that , for all , and is central valued on .

We would like to remark that the same conclusions hold in case we consider the right annihilator, more precisely.

Theorem B. Let be a prime ring of characteristic different from 2, with extended centroid , its two-sided Utumi quotient ring, a nonzero generalized derivation of , a noncentral multilinear polynomial over in n noncommuting variables, and such that for any . Then either or one of the following holds: (1)there exists such that , for all ;(2)there exist and such that , for all , and is central valued on .

Here we will consider a more general situation, involving a two-sided annihilating condition. More specifically, we study simultaneously left and right annihilators of the set and prove the following.

Theorem 1. Let be a prime ring of characteristic different from 2, with extended centroid , its two-sided Utumi quotient ring, a nonzero generalized derivation of , a noncentral multilinear polynomial over in n noncommuting variables, and such that for any . Then one of the following holds: (1);(2);(3)there exists such that , for all ;(4)there exist and such that , for all , and is central valued on ;(5)there exist and such that , for all , and , .

Remark 2. By the primeness of and in light of Theorems A and B, we may assume that is not a domain. Moreover, since the center of a prime ring cannot contain nonzero zero-divisor, then neither nor . Finally in all that follows we always suppose .

In the sequel we will make a frequent use of the following.

Remark 3. If is a basis of over then any element of , the free product over of the -algebra and the free -algebra , is called generalized polynomial and can be written in the form . In this decomposition the coefficients are in and the elements are -monomials; that is, , with and . In [12] it is shown that a generalized polynomial is the zero element of if and only if all are zero. Let be linearly independent over and , for some . If, for any , and , then are the zero element of . The same conclusion holds if , and for some .
We refer the reader to [1, 12] for more details on generalized polynomial identities.

2. An Independent Result

We will dedicate this section to the proof of the following proposition on linear identities with commutators in matrix rings. This result will be useful in the sequel.

Proposition 4. Let be a field and the algebra of matrices over and . Let , such that and for all . Then there exists such that and .

In order to prove Proposition 4, we need several lemmas.

Lemma 5. Let be an infinite field and . If are not scalar matrices in , then there exists some invertible matrix such that each matrix has all nonzero entries.

Proof. See Lemma  1.5 in [13].

Lemma 6. Let be a prime ring with extended centroid . Suppose , for all , where , for and . If are -independent then each is -dependent on . Analogously if are -independent then each is -dependent on .

Proof. It is Martindale’s result contained in [14].

Lemma 7. Let be a prime ring with extended centroid . Suppose , for all , where . Then .

Proof. It is an easy consequence of Lemma 6.

Lemma 8. Let be an infinite field, the algebra of matrices over , the center of , and . Assume that there exist nonzero elements of such that for all . If then one of the following holds: (1) are central matrices and ;(2) is a central matrix and .

Proof. Since , by the assumption, we have that for all . Clearly if then for all , and by Lemma 7 we get ; that is, . On the other hand, if , then for all and it follows easily that .
In light of this, we consider and both nonscalar matrices. We will prove that in this case we get a contradiction.
Here we denote by the usual matrix unit with in the -entry and zero elsewhere.
By Lemma 5, we can assume that and have all nonzero entries, say and , for , .
Since for all , then, for any , in particular the -entry of is , a contradiction.

For sake of clearness, we may write the previous lemma as follows.

Lemma 9. Let be an infinite field, the algebra of matrices over , the center of , and . Let be nonzero elements of such that for all . Assume there exists such that . Then and , for a suitable .

Lemma 10. Let be an infinite field, the algebra of matrices over , and the center of . Assume that there exist nonzero elements of such that for all . If and , for a suitable , then .

Proof. Assume that is not a scalar matrix. By Lemma 5, we can assume that and have all nonzero entries, say and , for , .
Since , for a suitable , by our assumption we have that that is, for all . In particular for , with , By calculations one has that the -entry of is , a contradiction.
Therefore must be a central matrix. In light of this, there exist such that and , so that , for all . Once again by Lemma 7 and since , it follows that ; that is, and .

Lemma 11. Let be an infinite field, the algebra of matrices over , and . Suppose there exist such that for all . Denote for suitable , and elements of . If there are such that , , and , then and for all and (i.e., the only nonzero off-diagonal elements of fall in the th row).

Proof. Consider the assumption In particular, for , we have so that, for all , the -entry of the matrix is . Since , one has for all , in particular . Thus, in case we are done (since ).
Assume in what follows that , and choose , with . Hence we also have From the previous equalities it follows that(1)for all , the -entry of the matrix is ;(2)for all , the -entry of the matrix is ;(3)the -entry of the matrix is ;(4)for all , the -entry of the matrix is (note that this holds also in case ).By and and since and , we have both , for all , and for all and . So by for all . Finally by , for all and .

Lemma 12. Let be an infinite field, the algebra of matrices over , and . Suppose there exist such that for all . Denote for suitable , , and elements of . Assume there are such that . If , for all , then one of the following holds: (1);(2), , and there exists such that

Proof. Firstly we consider the case . The first step is to apply twice Lemma 11: this forces to be a diagonal matrix. In fact , , and imply that for all and ; in particular, since , there exists such that , for all . Since , , we have for all and , so for all , as required. Say .
Consider now the inner automorphism of induced by the invertible matrix , for : . Of course , for all . Moreover the -entries of , , and are, respectively, , , and . Therefore, again by Lemma 11, any -entry of is zero, for all . By calculations ; that is, .
On the other hand, if is the inner automorphisms induced by the invertible matrix , as above , for all . Since the -entries of , , and are, respectively, , , and , and again any -entry of is zero, for all ; that is, and , for all . Thus is a central matrix in . By Lemma 9, either for some or . Since the first case cannot occur, we get and also which follows from and .
Let now ; that is, . In this case it is well known that for any element there exist such that . Without loss of generality we may assume . In case , then by the same above argument we show that and we are done again. Thus we consider the case . Moreover, by applying Lemma 11 it follows . Hence we may write For we have so that the -entry of the matrix is ; that is, and the -entry of the matrix is ; that is, . On the other hand, for , we have The -entry of the matrix is ; that is, and . Moreover the -entry of the matrix is . Therefore, if denoted , one has and .
Analogously, the -entry of the matrix is . Thus and .
Finally, by our assumption and for , with , we also have and by easy calculations it follows .

Lemma 13. Let be an infinite field, the algebra of matrices over , and . Let and denote for suitable , , , and elements of . Suppose and for all . Assume there are such that . If , , for all , then .

Proof. By our hypothesis, we have for all . By Lemma 12 it follows that either or and there exists such that Notice that implies that the following holds: Moreover, by computing the product we get Finally, by computing the product we also have Notice that, in case , by (20) it follows the contradiction . Thus and multiply (25) by , so that . Again by (20) we have and using (22) it follows . Since and , then .
Assume , denoted by the identity matrix in , and let .
Since and induce the same inner derivation, then by our assumptions we have that for all . By applying again Lemma 12, it follows that either or and there exists such that In the latter case, by using the same above argument, the matrix satisfies the equalities (22) and (25); that is, respectively, implying , and which is a contradiction.
Therefore In this case, by using both (22) and (30), the -entry of the matrix should be The previous contradiction implies ; that is, and by (26) also . Hence .
Now consider the following elements in : Thus and in particular the -entry of is and the -entry of is Since , then . Therefore the sum of (34) and (35) forces the contradiction .

Lemma 14. Let be an infinite field, the algebra of matrices over , and . Let and denote for suitable , , , and elements of . Suppose and for all . Then there exists such that and .

Proof. Clearly if one of , , , or is a scalar matrix we are done by Lemma 9. In order to prove this lemma, we may assume that , , , and are noncentral matrices.
By Lemma 5, there exists some invertible matrix such that , , , and have all nonzero entries.
Notice that are linearly -dependent if and only if are linearly -dependent; analogously are linearly -dependent if and only if are linearly -dependent. Moreover if and only if . Therefore, in order to prove our result, we may replace , respectively, by , so that , and have all nonzero entries.
For we have in particular the -entry of is . Denote , so that . Let be the identity matrix in and . Since and induce the same inner derivation in , then ; that is, , for all . Moreover and have all nonzero entries, and the -entry of is zero. Thus we may apply Lemmas 12 and 13 and obtain and , as required.

Proof of Proposition 4. If one assumes that is infinite, the conclusion follows from Lemma 14.
Now let be an infinite field which is an extension of the field and let . Consider the generalized polynomial which is a generalized polynomial identity for . Since is a multilinear generalized polynomial in the indeterminates , then it is a generalized polynomial identity for and the conclusion follows again from Lemma 14.

3. The Inner-Case in Prime Rings

In this section we consider , the set of all evaluations of the noncentral multilinear polynomial over , and assume that is an inner generalized derivation, so that there exist such that , for all , and satisfies where are nonzero elements of .

In order to prove the first result we premit the following.

Fact 1. Let be the algebra of matrices over of characteristic different from 2. Notice that the set is invariant under the action of all inner automorphisms of . Hence if denoted by , then for any inner automorphism of , we have that and .
Since is not central then, by [15] (see also [16]), there exist and , such that , with . Moreover, since the set is invariant under the action of all -automorphisms of , then for any there exist such that .

Now we may start with the following.

Proposition 15. Let be a field, the algebra of matrices over , and a noncentral multilinear polynomial over . Let and denote for suitable , , , and elements of . Suppose that for all . Then one of the following holds: (1);(2)there exists such that , and is central valued on ;(3)there exist such that , and .

Proof. By our assumption, satisfies the following generalized polynomial identity: As in the previous section denotes the matrix unit with in -entry and zero elsewhere.
Firstly we assume is an infinite field.
Since is not central then, by Fact 1, for any , there exist such that .
Then we obtain In particular, In light of Remark 2, we assume that and are not central matrices. Denote and suppose that is not scalar. By Lemma 5 there exists an -automorphism of such that , , and have all nonzero entries. Clearly , , and must satisfy the condition (44) and this is a contradiction.
This means that , for some , and the main condition is now for all ; that is, , for all .
Consider the additive subgroup of , generated by the set . By [17], either or the noncentral Lie ideal of is contained in . In the first case we conclude that is central valued in and we are done. In either case we have , for all , and by Proposition 4 we get the required conclusions.
Now let be an infinite field which is an extension of the field and let . Notice that the multilinear polynomial is central-valued on if and only if it is central-valued on . Consider the generalized polynomial which is a generalized polynomial identity for . Moreover it is multihomogeneous of multidegree in the indeterminates .
Hence the complete linearization of is a multilinear generalized polynomial in indeterminates; moreover Clearly the multilinear polynomial is a generalized polynomial identity for and too. Since we obtain , for all , and the conclusion follows from the argument contained in the first part of this proposition.

Lemma 16. If there exist , , such that , for all , then satisfies a nontrivial generalized polynomial identity, unless when one of the following holds: (1);(2) and there exists such that .

Proof. Assume that does not satisfy any nontrivial generalized polynomial identity with coefficients in . Therefore, is a trivial generalized polynomial identity for . By calculations for all . If and , the proof is completed; hence we suppose that and are not simultaneously central. By Remark 3 and by (49), if are linearly -independent then satisfies the trivial generalized polynomial identity . It means, since , , a contradiction. Analogously, if we suppose linearly -independent, we get , a contradiction.
Therefore there exist such that and ; now (49) becomes for all . Since it is a trivial generalized polynomial identity, then . Moreover, ; that is, .

Proposition 17. Let such that for all . Then one of the following holds: (1);(2)there exists such that , and is central valued on ;(3)there exist such that , , and .

Proof. By Remark 2 we assume that is not a domain.
Moreover, by Lemma 16, satisfies the nontrivial generalized polynomial identity: By a theorem due to Beidar (Theorem  2 in [18]) this generalized polynomial identity is also satisfied by . In case is infinite, we have for all , where is the algebraic closure of . Since both and are centrally closed [19, Theorems 2.5 and 3.5], we may replace by or according to being finite or infinite. Thus we may assume that is centrally closed over which is either finite or algebraically closed. By Martindale’s theorem [14], is a primitive ring having a nonzero socle with as the associated division ring, and is a simple central algebra finite dimensional over , for any minimal idempotent element .
In light of Jacobson’s theorem [20, page 75] is isomorphic to a dense ring of linear transformations on some vector space over .
Assume first that is finite-dimensional over . Then the density of on implies that , the ring of all matrices over . Since is not commutative we assume . In this case the conclusion follows by Proposition 15.
Assume next that is infinite-dimensional over . As in Lemma  2 in [21], the set is dense on and so from , for all , we have that satisfies the generalized identity . We remark that satisfies (see, e.g., [5, proof of Theorem 1]); that is, for all , In this equality we substitute with , for any nontrivial idempotent element , and obtain By the primeness of , it follows that either or or . Here our aim is to prove that in any case . To do this, we firstly assume that . In (53) replace by , so that , which implies .
Moreover we substitute in (53) with and by easy computation it follows ; that is, .
On the other hand, if one supposes and replacing in (53) by , one has , which implies . Finally, if substituted in (53) with , as above we have . Thus in any case it follows .
Similarly one can prove also that .
Hence , for any idempotent element . Since is not a domain, then is generated by its minimal idempotent elements; therefore ; that is, . Let such that . By our assumption it follows that satisfies that is satisfies . In this last replace by and obtain that satisfies . Since , then , for all . By [14, Lemma 1] it follows that there exists such that and , unless .

Corollary 18. Let such that and be a noncentral multilinear polynomial over . If , for all , then either or .

4. The Main Result

In [11] Lee proved that every generalized derivation can be uniquely extended to a generalized derivation of and thus all generalized derivations of will be implicitly assumed to be defined on the whole and obtained the following result.

Theorem 19 (Theorem  3 in [11]). Every generalized derivation on a dense right ideal of can be uniquely extended to and assumes the form , for some and a derivation on .

In this section we denote by the polynomial obtained from by replacing each coefficient with . Thus we write , for all in .

In light of this, we finally prove our main result.

Proof of Theorem 1. Suppose both and . Since satisfies the generalized differential identity the above cited Lee’s result says that satisfies If is an inner derivation induced by an element , then satisfies the generalized polynomial identity: which is In this case we are done by Proposition 17.
Hence let be an outer derivation of . In this case satisfies the differential identity: By Kharchenko’s theorem (see [16, 22]), satisfies the generalized polynomial identity: and in particular, for all , satisfies the blended component Let and replace any by . Thus satisfies that is, By Corollary 18, we get the contradiction .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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