Dependent Elements of Derivations on Semiprime Rings
We characterize dependent elements of a commuting derivation on a semiprime ring and investigate a decomposition of using dependent elements of . We show that there exist ideals and of such that is an essential ideal of , , on , , and acts freely on .
1. Introduction and Preliminaries
Murray and von Neumann  and von Neumann  introduced the notion of free action on abelian von Neumann algebras and used it for construction of certain factors. Kallman  generalized the notion of free action of automorphisms to von Neumann algebras, not necessarily abelian, by using implicitly the dependent elements of an automorphism. Dependent elements of automorphisms were later studied by Choda et al.  in the context of -algebras. Several other authors have studied dependent elements of automorphisms in the context of operator algebras (see [5, 6] and references therein). A brief account of dependent elements in -algebras has also appeared in the book of Strătilă .
It is well known that all and von Neumann algebras are semiprime rings; in particular a von Neumann algebra is prime if and only if its centre consists of the scalar multiples of identity . Thus a natural extension of the notion of a dependent element of mappings on a -algebra or von Neumann algebras is the study of this notion in the context of semiprime rings and prime rings.
Laradji and Thaheem  initiated the study of dependent elements of endomorphisms of semiprime rings and generalized a number of results of  for semiprime rings. Recently, Vukman and Kosi-Ulbl  and Vukman [11, 12] have made further study of dependent elements of some mappings on prime and semiprime rings.
On one hand, motivated by the work of Laradji and Thaheem , Vukman and Kosi-Ulbl , and Vukman [11, 12] on dependent elements of mappings of semiprime rings and on the other hand by the work done by various researchers on commuting derivations on prime and semiprime rings, we investigate some properties, not already investigated, of dependent elements of commuting derivations on semiprime rings. We show that the dependent elements of a commuting derivation of a semiprime ring are central and form a commutative semiprime subring of . We also show that for a commuting derivation on a semiprime ring , there exist ideals and of such that is an essential ideal of , , on , , and zero is the only dependent element of , the restriction of on ; that is, acts freely on .
Throughout, will represent an associative ring with centre . The commutator will be denoted by . We will use the basic commutator identities and . Recall that a ring is semiprime if implies and is prime if implies or . An additive mapping is called a derivation on if for all . It is called commuting if for all . Let , then the mapping given by is a derivation on . It is called inner derivation on .
We call an element element a dependent element of a derivation if holds for all . Following , a derivation is said to act freely on (or a free action) in case zero is the only dependent element of . It is known that a semiprime ring has no central nilpotent elements. We will use this fact without any specific reference. For a derivation , denotes the collection of all dependent elements of .
It is known that the left and right annihilators of an ideal of a semiprime ring coincide. It will be denoted by . It is also known that and is an essential ideal of . We will use these facts without any further reference.
We will use the following result in the sequel.
Theorem 1.1 (see [8,Corollary 3.2]). If is a commuting inner derivation on a semiprime ring , then .
We now prove our results.
Theorem 2.1. Let be a commuting derivation of a semiprimering . Then if and only if and for all .
Proof. Let . Then Replacing by in (2.1), we
get . That is, From (2.1) and (2.2), we get Multiplying (2.3) by on the right,
we get Replacing by in (2.3), we
get Subtracting (2.5) from (2.4), we get , which implies Multiplying (2.6) by on the left, we
get Replacing by in (2.6), we
get Subtracting (2.7) from (2.8), we get Since is commuting,
therefore from (2.9) we get From (2.10), we get Replacing by in (2.10) and
then subtracting the result from (2.11), we get , which implies . Using semiprimeness of , from the last relation we get Thus inner derivation defined by is commuting.
Hence by Theorem 1.1,
which implies . Thus . Further from (2.1), we get .
Conversely, let and let . Then . So . This completes the proof.
Corollary 2.2. Let be a semiprime ring and let be a commuting derivation of . Let , then .
Proof. Since , therefore Replacing by in (2.13), we get From (2.13), we get , which implies . Using (2.14), from the last relation we get Replacing by in (2.15) and using (2.15), we get . Thus for all . Using semiprimeness of , from the last equation we get .
Corollary 2.3. Let be a semiprime ring and let be a commuting derivation of . Then is a commutative semiprime subring of .
Proof. Let . Then by Theorem 2.1, , and for all . Obviously and . So and . Since , so . Thus is a commutative subring of . To show semiprimeness of , we consider , . Then for all . In particular , which implies because has no central nilpotents. Thus is a commutative semiprime subring of ring.
Corollary 2.4. Let be a commutative semiprime ring and let be a derivation of . Then is an ideal of .
Proof. Since is commutative, so is commuting. Let . Then by Corollary 2.3, . Let and let . Then for all . Thus . Since , so for all . Hence . Thus is an ideal of .
Remark 2.5. (i) If is a semiprime
ring and an ideal of , then it is easy to verify that is a semiprime
subring of and .
(ii) If is a commuting derivation on and , then by Theorem 2.1, for all . This implies , which gives . Thus for all , which by semiprimeness of implies .
Theorem 2.6. Let be a semiprime
ring and let be a commuting
derivation on . Then there exist ideals and of such that
(a) is an essential ideal of and ,(b) on and ,(c), where is restriction of on . That is, acts freely on .
Proof. (a) Let be the ideal of generated by . Let . Then is an ideal of , is an essential
ideal of , and .
(b) By Corollary 2.2 and Theorem 2.1, and for all and . By Remark 2.5(ii) . Thus , , and for and . Hence on .
Let . Thus for all . So , which implies . Thus because on . So, . Hence .
(c) Since is an ideal of , so by Remark 2.5(i) is a semiprime subring of and . Since , so is a derivation on . Let be a dependent element of , so by Theorem 2.1 and Corollary 2.2 , , and . Let , so . Thus , which implies . That is, , which implies . Since is an ideal of , so and for all . Replacing by in , we get . Using semiprimeness of , we get . Since and for all , therefore . So and . Thus . Hence . That is, acts freely on .
Since every derivation on a commutative ring is a commuting derivation and is an ideal of by Corollary 2.4, therefore in case of a commutative ring and . Thus we have the following corollary.
Corollary 2.7. Let be a
commutative semiprime ring and let be a derivation
on . Then there exist ideals and of such that
(a) is an essential ideal of and ,(b) on , ,(c), where is restriction of on . That is, acts freely on .
The authors are thankful to the referee for suggesting another proof of Theorem 2.6 with different ideals and . The ideal is generated by and . The proof suggested by the referee is based on a theorem of Chuang and Lee . The statement of the said theorem and the proof of Theorem 2.6 as suggested by the referee are given below.
Theorem 2.8 (see ). Let be a semiprime ring with a derivation and let be a left ideal of . Suppose that for all , where denotes the center of . Then .
Now we give the proof of Theorem 2.6 as suggested by the referee.
Proof. By Theorem 2.8, . Let be the ideal of generated by and let . Clearly, , , is essential in , and . Since , we have . Let be a dependent element of . Let . Then . Thus . Since is an essential ideal of the semiprime ring , we have , implying that . So , as asserted. This proves the theorem.
The authors are grateful for the support provided by Bahauddin Zakariya University, Multan, Pakistan.