Preprint
Article

Posner’s Theorem and *-Centralizing Derivations on Prime Ideals with Applications

Altmetrics

Downloads

146

Views

70

Comments

0

A peer-reviewed article of this preprint also exists.

Submitted:

02 June 2023

Posted:

02 June 2023

You are already at the latest version

Alerts
Abstract
A well-known result of Posner’s second theorem states that if the commutator of each element in a prime ring and its image under a nonzero derivation is central, then the ring is commutative. In the present paper, we extend this bluestocking theorem to an arbitrary ring with involution involving prime ideals. Further, apart from proving several other interesting and exciting results, we establish *-version of Vukman’s theorem [48, Theorem 1]. Precisely, we describe the structure of quotient ring A/L, where A is an arbitrary ring and L is a prime ideal of A. Further, by taking advantage of the *-version of Vukman’s theorem, we show that if a 2-trosion free semiprime ring A with involution admits a nonzero *-centralizing derivation, then A contains a nonzero central ideal. This result is in a spirit of the classical result due to Bell and Martindale [19, Theorem 3]. As the applications, we extends and unify several classical theorems proved in [6],[25,[42], and [48] . Finally, we conclude our paper with a direction for further research.
Keywords: 
Subject: Computer Science and Mathematics  -   Algebra and Number Theory

MSC:  Primary: 16N60, 16W10, Secondary:16W25

1. Introduction

The motivation for this paper lies in an attempt to extend in some way the famous results due to Posner [46], Vukman [48] and Ali-Dar [6]. A number of authors have generalized these theorems in several ways (see, for example, [16,20,33,35,39,42,50,51], where further references can be found). Throughout this article, A will represent an associative ring with center Z ( A ) . The standard polynomial identity s 4 in four variables is defined as s 4 ( 1 , 2 , 3 , 4 ) = σ s 4 ( 1 ) σ σ ( 1 ) σ ( 2 ) σ ( 3 ) σ ( 4 ) , where ( 1 ) σ is + 1 or 1 according to σ being an even or odd permutation in symmetric group s 4 . For any s , t A , the symbol [ s , t ] = s t t s stands for commutator, while the symbol s t will stand for the anti-commutator s t + t s . The higher order commutator is define as follows: for any s , t A ,
[ s , t ] 0 = s , [ s , t ] 1 = [ s , t ] = s t t s a n d [ s , t ] 2 = [ [ s , t ] , t ] ,
and inductively, we write [ s , t ] k = [ [ s , t ] k 1 , t ] , (where k > 1 is a fixed integer), is called commutator of order k or simply k t h -commutator. It is also known as Engel condition in literature (viz.; [35]). Analogously, we define the higher order anit-commutator, and set
s 0 t = s , s 1 t = s t + t s a n d s 2 t = ( s 1 t ) t ,
and inductively, we set s k t = ( s k 1 t ) t for k > 1 , is called anti-commutator of order k .
“Recall that a ring A is called prime if, for a , b A , a A b = ( 0 ) implies a = 0 or b = 0 . By a prime ideal of a ring A , we mean a proper ideal L and for , ϑ A , A y L implies that L or ϑ L . We note that for a prime ring A , ( 0 ) is the prime ideal of A and A / L is a prime ring. An ideal L of a ring A is called semiprime if it is the intersection of prime ideals or alternatively, if a A a L implies that a L for any a A . A ring A is said to be n-torsion free if n = 0 , A implies = 0 . An additive mapping * satisfying ( y ) * = ( ϑ ) * * and ( * ) * = is called an involution. A ring equipped with an involution is known as ring with involution or *-ring. An element in a ring with involution * is said to be hermitian if * = and skew-hermitian if * = . The sets of all hermitian and skew-hermitian elements of A will be denoted by H ( A ) and S ( A ) , respectively. If A is 2-torsion free then every A can be uniquely represented in the form 2 = h + k where h H ( A ) and k S ( A ) . The involution is said to be of the first kind if H ( A ) Z ( A ) , otherwise it is said to be of the second kind. We refer the reader to [32] for justification and amplification for the above mentioned notations and key definitions.
A map e : A A is a derivation of a ring A if e is additive and satisfies e ( ϑ ) = e ( ) ϑ + e ( ϑ ) for all , ϑ A . A derivation e is called inner if there exists a A such that e ( ) = [ a , ] for all A . An additive map F : A A is called a generalized derivation if there exists a derivation e of A such that F ( ϑ ) = F ( ) ϑ + e ( ϑ ) for all , ϑ A (see [14] for details). For a nonempty subset S of A , a mapping ξ : S A is called commuting (resp. centralizing) on S if [ ξ ( ) , ] = 0 (resp. [ ξ ( ) , ] Z ( A ) ) for all S . The study of commuting and centralizing mappings goes back to 1955 when Divinsky [31] proved that a simple artinian ring is commutative if it has a commuting automorphism different from the identity mapping. Two years later Posner [46] showed that a prime ring must be commutative if it admits a nonzero centralizing derivation. In 1970, Luh [38] generalized Divinsky’s result for prime rings. Later Mayne [44] established the analogous result of Posner for nonidentity centralizing automorphisms. The culminating results in this series can be found in [19,20,21,33,35,43] and [48,49]. In [48], Vukman generalized Posner’s second theorem for second order commutator and established that if a prime ring of characteristic different from 2 admits a nonzero derivation e such that [ e ( ) , ] 2 = 0 for all A , then A is commutative. Most classical and elegant generalization of Posner’s second theorem is due to Lanski [34]. Precisely, he proved that if a prime ring A admits a nonzero derivation e such that [ e ( ) , ] k = 0 for all L , where L is a non-commutative Lie ideal of A and k > 0 a fixed integer, then c h a r ( A ) = 2 and A M 2 ( F ) for a field F. These results have been extended in various ways (viz; [3,23,26,30,50,51] and references therein). The goal of this paper is to study these results in the setting of arbitrary rings with involution involving prime ideals and describe the structure of a quotient ring A / L , where A is an arbitrary ring and L is a prime ideal of A .
Let A be a ring with involution * and S be a nonempty subset of A . Following [6,25], a mapping ϕ of A into itself is called *-centralizing on S if ϕ ( ) * * ϕ ( ) Z ( A ) for all S , in the special case where ϕ ( ) * * ϕ ( ) = 0 for all S , the mapping ϕ is said to be *-commuting on S. In [6,25], the first author together with Dar initiated the study of these mappings and proved that the existence of a nonzero *-centralizing derivation of a prime ring with second kind involution forces the ring to be commutative. Apart from the characterizations of these mappings of prime and semiprime rings with involution, they also proved *-version of Posner’s second theorem and its related problems. Precisely, they established that : Let A be a prime ring with involution * such that c h a r ( A ) 2 . Let e be a nonzero derivation of A such that [ e ( ) , * ] Z ( A ) for all A and e ( S ( A ) Z ( A ) ) ( 0 ) . Then A is commutative. Futher, they showed that every * -commuting map f : A A on semiprine ring with involution of characteristic different from two is of the form f ( ) = λ * + μ ( ) for all A , λ C (the extended centroid of A ) and μ : A C is an additive mapping. In the sequel, recently Nejjar et al. [42, Theorem 3.7] established that if a 2-torsion free prime ring with involution of the second kind admits a nonzero derivation e such that e ( ) * * e ( ) Z ( A ) for all A , then A is commutative. In 2020, Alahmadi et al. [2] extend the above mentioned result for generalized derivations. Over the last few years the interest on this topic has been increased and numerous papers concerning these mappings on prime rings have been published (see [1,2,7,8,9,10,13,39,42,45] and references therein). In [24], Creedon studied the action of derivations of prime ideals and proved that if e is a derivation of a ring A and L is a semiprime ideal of A such that A / L is characteristic-free and e k ( L ) L , then e ( L ) L for some positive integer k.
In view of the above observations and motivation, the aims of the present paper is to prove the following main theorems.
Theorem-A Let A be a ring with involution * of the second kind and L a prime ideal of A such that S ( A ) Z ( A ) L . If e 1 and e 2 are derivations of A such that e 1 ( ) * * e 2 ( ) L for all A , then one of the following holds:
  • c h a r ( A / L ) = 2
  • e 1 ( A ) L and e 2 ( A ) L
  • A / L is a commutative integral domain.
Theorem-B Let A be a ring with involution * of the second kind and L a prime ideal of A such that S ( A ) Z ( A ) L . If e 1 and e 2 are derivations of A such that [ e 1 ( ) , * ] + [ , e 2 ( * ) ] + [ , * ] L for all A , then one of the following holds:
  • c h a r ( A / L ) = 2
  • A / L is a commutative integral domain.
Theorem-C Let A be a ring with involution * of the second kind and L a prime ideal of A such that S ( A ) Z ( A ) L . If A admits a derivation e such that e ( * ) e ( * ) e ( ) L for all A , then one of the following holds:
  • c h a r ( A / L ) = 2
  • e ( A ) L .
Theorem-D Let A be a ring with involution * of the second kind and L a prime ideal of A such that S ( A ) Z ( A ) L . If A admits a derivation e such that [ [ e ( ) , * ] , * ] L for all A , then one of the following holds:
  • c h a r ( A / L ) = 2
  • e ( A ) L
  • A / L is a commutative integral domain.
Theorem-E Let A be a 2-torsion free semiprime ring with involution * of the second kind. If A admits a nonzero *-centralizing derivation e , i . e . , [ e ( ) , * ] Z ( A ) for all A , then A contains a nonzero central ideal.
In view of * -centralizing mappings, Theorem-e and Theorem-E recognized as the * -versions of well known theorems due to Posner [46] and Vukman [48]. As the applications of Theorems A to E just mentioned above, we extends and unify several classical theorems proved in [6,7,25,42,46] and [48,49]. Since these results are in new direction, so there are various interesting open problems related to our work. Hence, we conclude our paper with a direction for further research in this new and exciting area of theory rings with involution.
We shall do a great deal of calculation with commutators and anti-commutators, routinely using the following basic identities: For all s , t , w A ;
[ s t , w ] = s [ t , w ] + [ s , w ] t a n d [ s , t w ] = t [ s , w ] + [ s , t ] w
s ( t w ) = ( s t ) w t [ s , w ] = t ( s w ) + [ s , t ] w
( s t ) w = s ( t w ) [ s , w ] t = ( s w ) t + s [ t , w ] .

2. Preliminary Results

Let A be * -ring. Following [8,47], an additive mapping e : R R is called a *-derivation of A if e ( ϑ ) = e ( ) ϑ * + d ( ϑ ) for all , ϑ A . An additive mapping e : A A is called a Jordan *-derivation of A if e ( 2 ) = e ( ) * + e ( ) for all A . In [21], Brešar showed that if a prime ring A admit nonzero derivations e 1 and e 2 of A such that e 1 ( ) e 2 ( ) Z ( A ) for all I , where I is a nonzero left ideal of A , then A is commutative. Further, this result was extended by Argac [12] as follows: Let A be a semiprime ring and e 1 , e 2 are derivations of A such that at least one is nonzero. If e 1 ( ) e 2 ( ) Z ( A ) for all A , then A contains a nonzero central ideal. Motivated by the above mentioned results, first author together with Alhazmi et al. [10] studied more general problem in the setting of rings with involution. Prescisely, they proved that if a ( m + n ) ! -torsion free prime ring with involution of the second kind admit Jordan *-derivations e and g of A such that e ( m ) n ± n g ( m ) = 0 for all A (where m and n are fixed positive integers), then e = g = 0 or A is commutative. In the sequel, very recently Nejjar et al. [42, Theorem 3.7] established that if a 2-torsion free prime ring with involution of the second kind admits a nonzero derivation e such that e ( ) ( ) * ( ) * e ( ) Z ( A ) for all A , then A is commutative. The goal of this section is to initiate the study of a more general concept than *-centralizing mappings are; that is, we consider the situation when the mappings ϕ and ξ of a ring A satisfy ϕ ( ) ( ) * ( ) * ξ ( ) L for all A , where A is an arbitrary ring and L is a prime ideal of A . Precisely, we prove the following theorem.
 Theorem 1. 
Let A be a ring with involution * of the second kind and L a prime ideal such that S ( A ) Z ( A ) L . If e 1 and e 2 are derivations of A such that e 1 ( ) ( ) * ( ) * e 2 ( ) L for all A , then one of the following holds:
  • c h a r ( A / L ) = 2
  • e 1 ( A ) L and e 2 ( A ) L
  • A / L is a commutative integral domain.
Following are the immediate consequences of Theorem 1. In fact, Corollary 1 is in sprit of the result due to Posner’s second theorem.
 Corollary 1. 
Let A be a ring with involution * of the second kind and L a prime ideal of A such that S ( A ) Z ( A ) L . If A admits a derivation e such that [ e ( ) , ( ) * ] L for all A , then one of the following holds:
  • c h a r ( A / L ) = 2
  • e ( A ) L
  • A / L is a commutative integral domain.
 Corollary 2. 
Let A be a ring with involution * of the second kind and L a prime ideal of A such that S ( A ) Z ( A ) L . If A admits a derivation e such that e ( ) ( ) * L for all A , then one of the following holds:
  • c h a r ( A / L ) = 2
  • e ( A ) L .
 Corollary 3. 
Let A be a prime ring with involution * of the second kind such that c h a r ( A ) 2 . If A admits a *-commuting derivation e, then e = 0 or A is a commutative integral domain.
 Corollary 4. 
Let A be a prime ring with involution * of the second kind such that c h a r ( A ) 2 . If A admits a derivation e such that e ( ) ( ) * = 0 for all A , then e = 0 .
For the proof of Theorem 1, we need the following lemmas, some of which are of independent interest. We begin our discussions with the following.
 Lemma 1. 
[5, Lemma 2] Let A be a ring, L be a prime ideal of A . If e is a derivation of A satisfying the condition [ e ( ) , ] L for all A , then e ( A ) L or A / L is a commutative.
 Lemma 2. 
[41, Lemma 1] Let A be a ring, L a prime ideal of A , e 1 and e 2 derivations of A . Then e 1 ( ) e 2 ( ) L for all A if and only if ( e 1 ( A ) L and e 2 ( A ) L ) or A / L is a commutative integral domain."
 Lemma 3. 
Let A be a ring with involution * of the second kind and L a prime ideal of A such that S ( A ) Z ( A ) L . If [ , ( ) * ] L for all A , then one of the following holds:
  • c h a r ( A / L ) = 2
  • A / L is a commutative integral domain.
 Proof. 
We assume that c h a r ( A / L ) 2 . By the assumption, we have
[ , ( ) * ] L
for all A . Direct linearization of relation (1) gives
[ , ϑ * ] + [ ϑ , ( ) * ] L
for all , ϑ A . Replacing by k in (2), where 0 k S ( A ) Z ( A ) , we get
k [ , ( ϑ ) * ] k [ ϑ , ( ) * ] L
for all , ϑ A . Since S ( A ) Z ( A ) L , it follows that
[ , ( ϑ ) * ] [ ϑ , ( ) * ] L
for all , ϑ A . Combining (2) and (3), we obtain
2 [ , ( ϑ ) * ] L
for all , ϑ A . This implies that
[ , ϑ ] L
for all , ϑ A . Since elements of A / L are cosets and notice that L implies + L = L . Therefore, the above equation gives
ϑ ϑ + L = L
for all , ϑ A and hence, we infer that
ϑ + L = ϑ + L
for all , ϑ A . This can be written as
( + L ) ( ϑ + L ) = ( ϑ + L ) ( + L )
for all , ϑ A . This implies that A / L is commutative. Now we show that A / L is integral domain. We suppose that
( + L ) ( ϑ + L ) = L
for all , ϑ A . This is equivalent to the expression
ϑ + L = L
for all , ϑ A . This implies that ϑ L for all , ϑ A . For any r A , we have r ( ϑ ) L for all , ϑ A . This gives r ϑ L . Hence, A ϑ L . Thus, we get L or ϑ L . This further implies that + L = L or ϑ + L = L . This shows that A / L is an integral domain. Consequently, we conclude that A / L is a commutative integral domain. This proves the lemma. □
In view of Lemmas 2.6 & 2.8 , we conclude the following result.
 Lemma 4. 
Let A be a ring, L be a prime ideal of A . If e is a derivation of A satisfying the condition [ e ( ) , ] L for all A , then e ( A ) L or A / L is a commutative integral domain.
We are now ready to prove our first main theorem.
Proof of Theorem 1. We assume that c h a r ( A / L ) 2 . By the assumption, we have
e 1 ( ) ( ) * ( ) * e 2 ( ) L f o r a l l A .
Linearizing (8), we have
e 1 ( ) ( ϑ ) * + e 1 ( ϑ ) ( ) * ( ) * e 2 ( ϑ ) ( ϑ ) * e 2 ( ) L f o r a l l , ϑ A .
Replacing by h in (9), where 0 h H ( A ) Z ( A ) , we get
( e 1 ( ) ( ϑ ) * + e 1 ( ϑ ) ( ) * ( ) * e 2 ( ϑ ) ( ϑ ) * e 2 ( ) ) h + ( ϑ ) * e 1 ( h ) ( ϑ ) * e 2 ( h ) L f o r a l l , ϑ A .
Application of (9) yields
( ϑ ) * e 1 ( h ) ( ϑ ) * e 2 ( h ) L f o r a l l , ϑ A .
This gives that
ϑ e 1 ( h ) ϑ e 2 ( h ) L f o r a l l , ϑ A .
Replace h by k 2 in (10), where 0 k S ( A ) Z ( A ) L , to get
ϑ e 1 ( k ) ϑ e 2 ( k ) L f o r a l l , ϑ A .
Substituting k in place of in (9), where 0 k S ( A ) Z ( A ) , we arrive at
e 1 ( ) ( ϑ ) * k + ( ϑ ) * e 1 ( k ) e 1 ( ϑ ) ( ) * k + ( ) * e 2 ( ϑ ) k ( ϑ ) * e 2 ( k ) ( ϑ ) * e 2 ( ) k L
for all , ϑ A . From (9), we have
e 1 ( ) ( ϑ ) * k + e 1 ( ϑ ) ( ) * k ( ) * e 2 ( ϑ ) k ( ϑ ) * e 2 ( ) k L f o r a l l , ϑ A .
Adding (12) and (13), we obtain
2 e 1 ( ) ( ϑ ) * k 2 ( ϑ ) * e 2 ( ) k + ( ϑ ) * e 1 ( k ) ( ϑ ) * e 2 ( k ) L f o r a l l , ϑ A
this impels
2 e 1 ( ) ϑ k 2 ϑ e 2 ( ) k + ϑ e 1 ( k ) ϑ e 2 ( k ) L
for all , ϑ A . Using (11) in (14), we have
2 e 1 ( ) ϑ k 2 ϑ e 2 ( ) k L f o r a l l , ϑ A .
Since c h a r ( A / L ) 2 and S ( A ) Z ( A ) L , we have
e 1 ( ) ϑ ϑ e 2 ( ) L f o r a l l , ϑ A .
In particular, for ϑ = , we get e 1 ( ) e 2 ( ) L for all A . Therefore, from Lemma 2, we conclude that ( e 1 ( A ) L and e 2 ( A ) L ) or A / L is a commutative integral domain.□
 Corollary 5. 
Let A be a prime ring with involution * of the second kind such that c h a r ( A ) 2 . If A admit derivations e 1 and e 2 such that e 1 ( ) ( ) * ( ) * e 2 ( ) = 0 for all A , then e 1 = e 2 = 0 or A is a commutative integral domain.
We now prove another theorem in this vein.
 Theorem 2. 
Let A be a ring with involution * of the second kind and L a prime ideal of A such that S ( A ) Z ( A ) L . If A admits a derivation e such that [ e ( ) , ( ) * ] + [ , ( ) * ] L for all A , then one of the following holds:
  • c h a r ( A / L ) = 2
  • A / L is a commutative integral domain.
 Proof. 
Suppose that c h a r ( A / L ) 2 . By the assumption, we have
[ e ( ) , ( ) * ] + [ , ( ) * ] L
for all A . First we assume that e ( A ) L . Then, result follows by Lemma 3. Henceforward, we suppose that e ( A ) L . Linearizing (16), we get
[ e ( ) , ( ϑ ) * ] + [ d ( ϑ ) , ( ) * ] + [ , ( ϑ ) * ] + [ ϑ , ( ) * ] L
for all , ϑ A . Replacing by h in (17), where 0 h H ( A ) Z ( A ) , we get
e ( h ) [ , ( ϑ ) * ] L
for all , ϑ A . Replacing h by k 2 in the last relation, where 0 k S ( A ) Z ( A ) L and using the hypotheses, we arrive at
e ( k ) [ , ( ϑ ) * ] L
for all , ϑ A . Replacing by k in (17), where 0 k S ( A ) Z ( A ) L , we find that
d ( k ) [ , ( ϑ ) * ] + k [ e ( ) , ( ϑ ) * ] k [ d ( ϑ ) , ( ) * ] + k [ , ( ϑ ) * ] k [ ϑ , ( ) * ] L
for all , ϑ A . Using (18) and the condition S ( A ) Z ( A ) L in (19), we obtain
[ e ( ) , ( ϑ ) * ] [ d ( ϑ ) , ( ) * ] + [ , ( ϑ ) * ] [ ϑ , ( ) * ] L
for all , ϑ A . Addition of (17) and (20) gives that
2 ( [ e ( ) , ( ϑ ) * ] + [ , ( ϑ ) * ] ) L
for all , ϑ A . This implies
[ e ( ) , ϑ ] + [ , ϑ ] L
for all , ϑ A . In particular, for ϑ = , we have
[ e ( ) , ] L
for all A . In view of Lemma 4, we conclude that A / L is a commutative integral domain. □
The following result is interesting in itself.
 Theorem 3. 
Let A be a ring with involution * of the second kind and L a prime ideal of A such that S ( A ) Z ( A ) L . If e 1 and e 2 are derivations of A such that [ e 1 ( ) , ( ) * ] + [ , e 2 ( ( ) * ) ] + [ , ( ) * ] L for all A , then one of the following holds:
  • c h a r ( A / L ) = 2
  • A / L is a commutative integral domain.
 Proof. 
Assume that c h a r ( A / L ) 2 . By the assumption, we have
[ e 1 ( ) , ( ) * ] + [ , e 2 ( ( ) * ) ] + [ , ( ) * ] L
for all A . We divide the proof in three cases.
Case (i): First we assume that e 2 ( A ) L . Then, relation (21) reduces to
[ e 1 ( ) , ( ) * ] + [ , ( ) * ] L
for all A . In view of Theorem 2, we get the required result.
Case (ii): Now we assume that e 1 ( A ) L . Then, relation (21) reduces to
[ , e 2 ( ( ) * ) ] + [ , ( ) * ] L
for all A . This can be further written as
[ e 2 ( ( ) * ) , ] + [ ( ) * , ] L
for all A . If e 2 ( A ) L , then result follows by Lemma 3. Henceforward, we suppose that e 2 ( A ) L . Linearizing (22), we get
[ e 2 ( ( ) * ) , ϑ ] + [ e 2 ( ( ϑ ) * ) , ] + [ ( ) * , ϑ ] + [ ( ϑ ) * , ] L
for all , ϑ A . Replacing by h in (23), where 0 h H ( A ) Z ( A ) , we get
e 2 ( h ) [ ( ) * , ϑ ] L
for all , ϑ A . This implies that
e 2 ( h ) [ , ϑ ] L
for all , ϑ A . Replacing h by k 2 in the last relation, where 0 k S ( A ) Z ( A ) , we arrive at
2 e 2 ( k ) [ , ϑ ] k L
for all , ϑ A . Since c h a r ( A / L ) 2 , the last relation gives
e 2 ( k ) [ , ϑ ] k L
for all , ϑ A . Replacing by k in (23), where 0 k S ( A ) Z ( A ) , we find that
e 2 ( k ) [ ( ) * , ϑ ] k [ e 2 ( ( ) * ) , ϑ ] + k [ e 2 ( ( ϑ ) * ) , ] k [ ( ) * , ϑ ] + k [ ( ϑ ) * , ] L
for all , ϑ A . Left multiplying in (23) by k, we obtain
k [ e 2 ( ( ) * ) , ϑ ] + k [ e 2 ( ( ϑ ) * ) , ] + k [ ( ) * , ϑ ] + k [ ( ϑ ) * , ] L
for all , ϑ A . Combining (25) and (26), we get
e 2 ( k ) [ ( ) * , ϑ ] + 2 k [ e 2 ( ( ϑ ) * ) , ] + 2 k [ ( ϑ ) * , ] L
for all , ϑ A . Replacing ϑ by ϑ k in (27) and using (24), where 0 k S ( A ) Z ( A ) , we get
2 k 2 ( [ e 2 ( ϑ ) , ] + [ ϑ , ] ) L
for all , ϑ A . Using the assumption c h a r ( A / L ) 2 , we find that
k 2 ( [ e 2 ( ϑ ) , ] + [ ϑ , ] ) L
for all , ϑ A . Application of the primeness of L yields k 2 L or [ e 2 ( ϑ ) , ] + [ ϑ , ] L . The first case k 2 L implies k L , which gives a contradiction. Thus, we have
[ e 2 ( ϑ ) , ] + [ ϑ , ] L
for all , ϑ A . In particular for ϑ = , we have [ e 2 ( ) , ] L for all A . Therefore, in view of Lemma 4, we conclude that A / L is a commutative integral domain.
Case (iii): Finally, we assume that e 1 ( A ) L and e 2 ( A ) L . Then direct linearization of (21) gives
[ e 1 ( ) , ( ϑ ) * ] + [ e 1 ( ϑ ) , ( ) * ] + [ , e 2 ( ( ϑ ) * ) ] + [ ϑ , e 2 ( ( ) * ) ] + [ , ( ϑ ) * ] + [ ϑ , ( ) * ] L
for all , ϑ A . Replacing by h in (29), where 0 h H ( A ) Z ( A ) and using it, we get
e 1 ( h ) [ , ( ϑ ) * ] + e 2 ( h ) [ ϑ , ( ) * ] L
for all , ϑ A . Replacing by k in (30), where 0 k S ( A ) Z ( A ) L , we obtain
e 1 ( h ) [ , ( ϑ ) * ] e 2 ( h ) [ ϑ , ( ) * ] L
for all , ϑ A . Combination of (30) and (31) yields that
2 e 1 ( h ) [ , ( ϑ ) * ] L f o r a l l , ϑ A ,
which implies
e 1 ( h ) [ , ϑ ] L f o r a l l , ϑ A .
Replacing h by k 2 in the last relation and using the hypotheses of theorem, we get
e 1 ( k ) [ , ϑ ] L f o r a l l , ϑ A .
This implies either e 1 ( k ) L or [ , ϑ ] L . If [ , ϑ ] L , then by Lemma 3, A / L is a commutative integral domain. On the other hand, we have e 1 ( k ) L . Similarly, we can find e 2 ( k ) L . Writing k instead of in (29), where 0 k S ( A ) Z ( A ) L and using the fact that e 1 ( k ) , e 2 ( k ) L , we arrive at
[ e 1 ( ) , ( ϑ ) * ] [ e 1 ( ϑ ) , ( ) * ] + [ , e 2 ( ( ϑ ) * ) ] [ ϑ , e 2 ( ( ) * ) ] + [ , ( ϑ ) * ] [ ϑ , ( ) * ] L
for all , ϑ A . Comparing (29) and (32), we obtain
2 ( [ e 1 ( ) , ( ϑ ) * ] + [ , e 2 ( ( ϑ ) * ) ] + [ , ( ϑ ) * ] ) L f o r a l l , ϑ A .
This implies
[ e 1 ( ) , ϑ ] + [ , e 2 ( ϑ ) ] + [ , ϑ ] L f o r a l l , ϑ A .
Now, replacing by r in the above expression, we obtain
e 1 ( ) [ r , ϑ ] + [ , ϑ ] e 1 ( A ) L f o r a l l , ϑ , r A .
In particular, for ϑ = we have e 1 ( ) [ r , ] ` ` L for all , r A . This gives e 1 ( ) A [ r , ] L for all , r A . The primeness of L infers that e 1 ( ) L or [ r , ] L . Set A = { A | e 1 ( ) L } and A = { A | [ r , ] L } . Clearly, A and B are additive subgroups of A such that A B = A . But, a group cannot be written as a union of its two proper subgroups, consequently A = A or B = A . The first case contradicts our suppostion that e 1 ( A ) P . Thus, we have [ r , ] L for all r , A . Therefore, in view of Lemma 3, A / L is a commutative integral domain. This completes the proof of theorem. □
Using similar approach with necessary variations, one can establish the following result.
 Theorem 4. 
Let A be a ring with involution * of the second kind and L a prime ideal of A such that S ( A ) Z ( A ) L . If e 1 and e 2 are derivations of A such that [ e 1 ( ) , ( ) * ] + [ , e 2 ( ( ) * ) ] [ , ( ) * ] L for all A , then one of the following holds:
  • c h a r ( A / L ) = 2
  • A / L is a commutative integral domain.
In view of Theorems 3 & 4, we have the following corollaries:
 Corollary 6. 
Let A be a prime ring with involution * of the second kind such that c h a r ( A ) 2 . If A admit derivations e 1 and e 2 such that [ e 1 ( ) , ( ) * ] + [ , e 2 ( ( ) * ) ] ± [ , ( ) * ] = 0 for all A , then A is a commutative integral domain.
 Corollary 7. 
Let A be a ring with involution * of the second kind and L a prime ideal of A such that S ( A ) Z ( A ) ) L . If A admits a derivation e such that e ( [ , ( ) * ] ) ± [ , ( ) * ] L for all A , then one of the following holds:
  • c h a r ( A / L ) = 2
  • A / L is a commutative integral domain.
 Corollary 8. 
Let A be a ring with involution * of the second kind and L a prime ideal of A such that S ( A ) Z ( A ) ) L . If A admits a derivation e such that e ( ( ) * ) ± ( ) * L for all A , then one of the following holds:
  • c h a r ( A / L ) = 2
  • A / L is a commutative integral domain.
 Corollary 9 
([25], Theorem 3.4 ). Let A be a prime ring with involution * of the second kind such that c h a r ( A ) 2 . If A admits a derivation e such that e ( [ , ( ) * ] ) ± [ , ( ) * ] = 0 for all A , then A is a commutative integral domain.
We leave the open question whether or not the assumption S ( A ) Z ( A ) L (where L is prime ideal of an arbitray ring A ) can be removed in Theorems 1 and 3. In view of Theorem 1 and Theorem 4.4 of [10], we conclude this section with the following conjecture.
Conjecture: Let m and n be fixed positive integers. Next, let A be a * -ring with suitable torsion restrictions and L be a prime ideal of A . If A admit Jordan *-derivations e and g of A such that e ( ) ( m ) ( ) * n ± ( ) * n g ( m ) L for all A . Then, what we can say about the structure of A and the forms of e , g ?

3. Derivations act as homomorphisms and anti-homomorphisms on prime ideals

Ring homomorphisms are mappings between two rings that preserve both addition and multiplication. In particular, we are concerned with ring homomorphisms between two rings. If A is the real number field, then the zero map and the identity are typical examples of ring homomorphisms on A . Let S be a nonempty subset of A and e a derivation on A . If e ( ϑ ) = e ( ) e ( ϑ ) or e ( ϑ ) = e ( ϑ ) e ( ) for all , ϑ S , then e is said to be a derivation which acts as a homomorphism or an anti-homomorphism on S, respectively. Of course, derivations which acts as an endomorphisms or anti-endomorphisms of a ring A may behave as such on certain subsets of A , for example, any derivation e behaves as the zero endomorphism on the subring T consisting of all constants (i.e., elements for which e ( ) = 0 ). In fact, in a semiprime ring A , e may behave as an endomorphism on a proper ideal of A . As an example of such A and e, let S be any semiprime ring with a nonzero derivation δ , take A = S S and define e by e ( r 1 , r 2 ) = ( δ ( r 1 ) , 0 ) . However in case of prime rings, Bell and Kappe [18] showed that the behaviour of e is some what more restricted. By proving that if A is a prime ring and e is a derivation of A which acts as a homomorphism or an anti-homomorphism on a nonzero right ideal of A , then e = 0 on A . Further, Ali et al. obtained [4] the above mentioned result for Lie ideals. Recently, Mamouni et al. [40] studied the above mentioned problem for prime ideals of an arbitrary ring by cosidering the identity e ( ϑ ) e ( ) e ( ϑ ) L for all , ϑ A or e ( ϑ ) e ( ϑ ) e ( ) L for all , ϑ A , where L is prime ideal of A . In the present section, our objective is to extend the above study in the setting of rings with involution involving prime ideals. In fact, we prove the following result:
 Theorem 5. 
Let A be a ring with involution * of the second kind and L a prime ideal of A such that S ( A ) Z ( A ) L . If A admits a derivation e such that e ( ( ) * ) e ( ( ) * ) e ( ) L for all A , then one of the following holds:
  • c h a r ( A / L ) = 2
  • e ( A ) L .
 Proof. 
Assume that c h a r ( A / L ) 2 . By the hypothesis, we have
e ( ( ) * ) e ( ( ) * ) e ( ) L
for all A . Linearization of (33) gives that
e ( ( ϑ ) * ) + e ( ϑ ( ) * ) e ( ( ) * ) e ( ϑ ) e ( ( ϑ ) * ) e ( ) L
for all , ϑ A . Replacing by h in (34), where 0 h H ( A ) Z ( A ) , we get
e ( h ) ( ( ϑ ) * + ϑ ( ) * ( ) * e ( ϑ ) e ( ( ϑ ) * ) ) L
for all , ϑ A . Taking h = k 2 in (35), where 0 k S ( A ) Z ( A ) L and using the hypotheses of theorem, we obtain
e ( k ) ( ( ϑ ) * + ϑ ( ) * ( ) * e ( ϑ ) e ( ( ϑ ) * ) ) L
for all , ϑ A . Invoking the primeness of A yields that e ( k ) L or ( ϑ ) * + ϑ ( ) * ( ) * e ( ϑ ) e ( ( ϑ ) * ) ) L for all , ϑ A . Consider the case when
( ϑ ) * + ϑ ( ) * ( ) * e ( ϑ ) e ( ( ϑ ) * ) L
for all , ϑ A . Replacing by k in (36) and combining with the obtained relation, we get
2 ( ( ϑ ) * e ( ( ϑ ) * ) ) L
for all , ϑ A . This implies that
ϑ e ( ϑ ) L
for all , ϑ A . In particular for = k , where k S ( A ) Z ( A ) , we have ϑ e ( ϑ ) L for all ϑ A . Substituting ϑ r for ϑ in the last relation, we obtain ϑ e ( A ) L . This yields e ( A ) A e ( A ) L for all r A . Since L is a prime ideal of A , we have e ( A ) L . On the other hand, consider the case e ( k ) L . Replacing by k in (34), where 0 k S ( A ) Z ( A ) L , we get
e ( ( ϑ ) * ) e ( ϑ ( ) * ) + e ( ( ) * ) d ( ϑ ) e ( ( ϑ ) * ) e ( ) L
Combination of (34) and (38) gives that
2 ( e ( ( ϑ ) * ) e ( ( ϑ ) * ) e ( ) ) L
for all , ϑ A . This implies that
e ( ϑ ) e ( ϑ ) e ( ) L
for all , ϑ A . Taking ϑ = k in the above relation and using e ( k ) L , we get k e ( ) L for all A . Since S ( A ) Z ( A ) L , one can conclude that e ( A ) L . □
Applying an analogous argument, we have the following result.
 Theorem 6. 
Let A be a ring with involution * of the second kind and L a prime ideal of A such that S ( A ) Z ( A ) ) L . If A admits a derivation e such that e ( ( ) * ) e ( ) e ( ( ) * ) L for all A , then one of the following holds:
  • c h a r ( A / L ) = 2
  • e ( A ) L .
 Corollary 10. 
Let A be a prime ring with involution * of the second kind such that c h a r ( A ) 2 . If A admits a derivation e such that e ( ( ) * ) = e ( ( ) * ) e ( ) or e ( ( ) * ) = e ( ) e ( ( ) * ) for all A , then e = 0 .
 Theorem 7. 
Let A be a 2-torsion free semiprime ring with involution * of the second kind. If A admits a derivation e such that e ( ( ) * ) e ( ( ) * ) e ( ) = 0 for all A , then e = 0 .
 Proof. 
Assume that c h a r ( A / L ) 2 . By the assumption, we have
e ( ( ) * ) e ( ( ) * ) e ( ) = 0 f o r a l l A .
By the semiprimeness of A , there exists a family L = { L α : α } of prime ideals such that α L α = ( 0 ) (see [11] for details). For each L α in L , we have
e ( ( ) * ) e ( ( ) * ) e ( ) L α f o r a l l A .
Invoking Theorem 5, we conclude that e ( A ) L α . Consequently, we get e ( A ) α L α = ( 0 ) and hence result follows. Thereby the proof is completed. □
Analogusly, we can prove the following result.
 Theorem 8. 
Let A be a 2-torsion free semiprime ring with involution * of the second kind. If A admits a derivation e such that e ( ( ) * ) e ( ) e ( ( ) * ) = 0 for all A , then e = 0 .

4. Applications

In this section, we present some applications of the results proved in Section 2 . Vukman [48, Theorem 1] generalizes the classical result due to Posner (Posner’s second theorem) [46] and proved that if e is a derivation of a prime ring A of characteristic different from 2, such that [ [ e ( ) , ] , ] = [ e ( ) , ] 2 = 0 for all A , then e = 0 or A is commutative. In fact, in view of Posner’s second theorem, he merely showed that e is commuting, that is, [ e ( ) , ] = 0 for all A . In [29], Deng and Bell extended the above mentioned result for semiprime ring and established that if a 6-torsion free semiprime ring admits a derivation e such that [ [ e ( ) , ] , ] = 0 for all I with e ( I ) ( 0 ) where I is a nonzero left ideal of A , then A contains a nonzero central ideal. These results were further refined and extended by a number of algebraists (see for example, [3,23,26,30,33,36] and [50]). It is our aim in this section to study and extend Vukman’s and Posner’s results for arbitrary rings with involution involving prime ideals. In fact, we prove the *-versions of these theorems. Moreover, our approach is somewhat different from those employed by other authors. Precisely, we prove the following result.
 Theorem 9. 
Let A be a ring with involution * of the second kind and L a prime ideal of A such that S ( A ) Z ( A ) L . If A admits a derivation e such that [ [ e ( ) , ( ) * ] , ( ) * ] L for all A , then one of the following holds:
  • c h a r ( A / L ) = 2
  • e ( A ) L
  • A / L is a commutative integral domain."
A derivation e : A A is said to be * -centralizing if [ e ( ) , ( ) * ] Z ( A ) for all A . The last expression can be written as [ [ e ( ) , ( ) * ] , ( ) * ] = [ e ( ) , ( ) * ] 2 = 0 for all A . Consequently, Theorem 9 regarded as the * -version of Vukman’s theorem [48]. Applying Theorem 9, we also prove that if a 2-torsion free semiprime ring A with involution * of the second kind admiting a nonzero *-centralizing derivation, then A must contains a nonzero central ideal. In fact, we prove the following result.
 Theorem 10. 
Let A be a 2-torsion free semiprime ring with involution * of the second kind. If A admits a nonzero *-centralizing derivation e , i . e . , [ e ( ) , ( ) * ] Z ( A ) for all A , then A contains a nonzero central ideal.
As an immediate consequence of Theorem 10, we obtain the following result.
 Corollary 11. 
Let A be a 2-torsion free semiprime ring with involution * of the second kind. If A admits a nonzero *-commuting derivation e , i . e . , [ e ( ) , ( ) * ] = 0 for all A , then A contains a nonzero central ideal.
In order to prove of Theorem 10, we need the proof of Theorem 9.
Proof of Theorem 9. Assume that c h a r ( A / L ) 2 . By the hypothesis, we have
[ [ e ( ) , ( ) * ] , ( ) * ] L f o r a l l A .
A linearization of (39) yields that
[ [ e ( ) , ( ϑ ) * ] , ( ϑ ) * ] + [ [ e ( ) , ( ϑ ) * ] , ( ) * ] + [ [ e ( ϑ ) , ( ϑ ) * ] , ( ) * ] + [ [ e ( ϑ ) , ( ) * ] , ( ) * ] + [ [ e ( ) , ( ) * ] , ( ϑ ) * ] + [ [ e ( ϑ ) , ( ) * ] , ( ϑ ) * ] L
for all , ϑ A . Putting = in (40), we get
[ [ e ( ) , ( ϑ ) * ] , ( ϑ ) * ] + [ [ e ( ) , ( ϑ ) * ] , ( ) * ] [ [ e ( ϑ ) , ( ϑ ) * ] , ( ) * ] + [ [ e ( ϑ ) , ( ) * ] , ( ) * ] + [ [ e ( ) , ( ) * ] , ( ϑ ) * ] [ [ e ( ϑ ) , ( ) * ] , ( ϑ ) * ] L
for all , ϑ A . Combining (40) and (41), we obtain
[ [ e ( ) , ( ϑ ) * ] , ( ) * ] + [ [ e ( ϑ ) , ( ) * ] , ( ) * ] + [ [ e ( ) , ( ) * ] , ( ϑ ) * ] L
for all , ϑ A . Replacing ϑ by ϑ h in (42), where h H ( A ) Z ( A ) , we deduce that
e ( h ) [ [ ϑ , ( ) * ] , ( ) * ] L f o r a l l , ϑ A .
Taking h = k 2 , where k S ( A ) Z ( A ) L and using the hypothesis, we have
e ( k ) [ [ ϑ , ( ) * ] , ( ) * ] L f o r a l l , ϑ A .
Now, substituting ϑ k in place of ϑ in (42), where k S ( A ) Z ( A ) , we get
k [ [ e ( ) , ( ϑ ) * ] , ( ) * ] + k [ [ e ( ϑ ) , ( ) * ] , ( ) * ] + e ( k [ [ ϑ , ( ) * ] , ( ) * ] ) k [ [ e ( ) , ( ) * ] , ( ϑ ) * ] L
for all , ϑ A . Application of (43) and the condition S ( A ) Z ( A ) L yields
[ [ e ( ) , ( ϑ ) * ] , ( ) * ] + [ [ e ( ϑ ) , ( ) * ] , ( ) * ] [ [ e ( ) , ( ) * ] , ( ϑ ) * ] L
for all , ϑ A . From (42) and (44), we can obtain
2 ( [ [ e ( ϑ ) , ( ) * ] , ( ) * ] ) L f o r a l l , ϑ A .
This implies
[ [ e ( ϑ ) , ] , ] L f o r a l l , ϑ A .
Writing + z instead of , we get
[ [ e ( ϑ ) , ] , z ] + [ [ e ( ϑ ) , z ] , ] L f o r a l l , ϑ , z A .
Replacing z by z r in (45), we find that
[ z , ] [ e ( ϑ ) , r ] + [ e ( ϑ ) , z ] [ r , ] L f o r a l l r , , ϑ , z A .
In particular for r = , we have
[ z , ] [ e ( ϑ ) , ] L f o r a l l , ϑ , z A .
This gives
[ z , ] A [ e ( ϑ ) , ] L f o r a l l , ϑ , z A .
Since L is a prime ideal of A , we have [ z , ] L for all z A or [ e ( ϑ ) , ] L for all ϑ A . Let us set A = { A | [ , z ] L } and B = { A | [ e ( ϑ ) , ] L } . Clearly, A and B are additive subgroups of A whose union is A . But a group cannot be written as a union of its two proper subgroups, it follows that either A = A or B = A . In the first case, A / L is a commutative integral domain from Lemma 3. On the other hand, if [ e ( ϑ ) , ] L for all , ϑ A , then we get [ e ( ) , ( ) * ] L for all A . Hence, in view of Corollary 1, we conclude that e ( A ) L or A / L is a commutative integral domain. This completes the proof of theorem. □
Proof of Theorem 10. We are given that e : A A is *-centralizing derivation, that is, [ e ( ) , ( ) * ] Z ( A ) for all A . This implies that [ [ e ( ) , ( ) * ] , ϑ ] = 0 for all , ϑ A . This gives
[ [ e ( ) , ( ) * ] , ( ) * ] = 0 f o r a l l A .
In view of semiprimeness of A , there exists a family P = { L α : α } of prime ideals such that α L α = ( 0 ) (see [11] for more details). Let L denote a fixed one of the L α . Thus, we have
[ [ e ( ) , ( ) * ] , ( ) * ] L f o r a l l A a n d f o r a l l L L .
From the proof of Theorem 9, we observe that for each , either
[ z , ] L f o r a l l z A ( I )
or
[ e ( ϑ ) , ] L f o r a l l ϑ A ( I I )
Define A I to be the set of z A for which ( I ) holds and A I I the set of ϑ A for which ( I I ) holds. Note that both are additive subgroups of A and their union is equal to A. Thus either A I = A or A I I = A , and hence L satisfies one of the following:
[ z , ] L f o r a l l , z A ( I )
or
[ e ( ϑ ) , ] L f o r a l l , ϑ A ( I I )
Call a prime ideal in P a type-one prime if it satisfies ( I ) , and call all other members of P type-two primes. Define L 1 and L 2 respectively as the intersection of all type-one primes and the intersection of all type-two primes, and note that
L 1 L 2 = L 2 L 1 = L 1 L 2 = { 0 } .
Clearly, from both the cases, we can conclude that [ e ( ) , ] L for all A for all L T . This implies that [ e ( ) , ] L T L = { 0 } for all A . That is, [ e ( ) , ] = 0 for all A . Hence, in view of [19], A contains a nonzero central ideal.□
Jordan product version of Theorem 9 is the following.
 Theorem 11. 
Let A be a ring with involution * of the second kind and L a prime ideal of A such that S ( A ) Z ( A ) L . If A admits a derivation e such that ( e ( ) ( ) * ) ( ) * L for all A , then one of the following holds:
  • c h a r ( A / L ) = 2
  • e ( A ) L .
 Proof. 
Assume that c h a r ( A / L ) 2 . By the hypothesis, we have
( e ( ) ( ) * ) ( ) * L f o r a l l A .
A linearization of (47) yields that
( e ( ) ( ϑ ) * ) ( ϑ ) * + ( e ( ) ( ϑ ) * ) ( ) * + ( e ( ϑ ) ( ϑ ) * ) ( ) * + ( e ( ϑ ) ( ) * ) ( ) * + ( e ( ) ( ) * ) ( ϑ ) * + ( e ( ϑ ) ( ) * ) ( ϑ ) * L
for all , ϑ A . Putting = in (48), we get
( ( e ( ) ( ϑ ) * ) ( ϑ ) * ) + ( ( e ( ) ( ϑ ) * ) ( ) * ) ( ( e ( ϑ ) ( ϑ ) * ) ( ) * ) + ( ( e ( ϑ ) ( ) * ) ( ) * ) + ( ( e ( ) ( ) * ) ( ϑ ) * ) ( ( e ( ϑ ) ( ) * ) ( ϑ ) * ) L
for all , ϑ A . Combining (48) and (49), we obtain
( e ( ) ( ϑ ) * ) ( ) * + ( e ( ϑ ) ( ) * ) ( ) * + ( e ( ) ( ) * ) ( ϑ ) * L
for all , ϑ A . Substitution of ϑ h for ϑ in (50), where h H ( A ) Z ( A ) produce that
e ( h ) ( ( ϑ ( ) * ) ( ) * ) L f o r a l l , ϑ A .
Taking h = k 2 , where k S ( A ) Z ( A ) L and using the hypothesis, we have
e ( k ) ( ( ϑ ( ) * ) ( ) * ) L f o r a l l , ϑ A .
Next, substitute ϑ k in place of ϑ in (50), where k S ( A ) Z ( A ) , we get
k ( ( e ( ) ( ϑ ) * ) ( ) * ) + k ( ( e ( ϑ ) ( ) * ) ( ) * ) + e ( k ) ( ( ϑ ( ) * ) ( ) * ) ) k ( ( e ( ) ( ) * ) ( ϑ ) * ) L
for all , ϑ A . Application of (51) and the condition S ( A ) Z ( A ) L yields
( ( e ( ) ( ϑ ) * ) ( ) * ) + ( ( e ( ϑ ) ( ) * ) ( ) * ) ( ( e ( ) ( ) * ) ( ϑ ) * ) L
for all , ϑ A . From (50) and (52), we can obtain
2 ( ( e ( ϑ ) ( ) * ) ( ) * ) L f o r a l l , ϑ A .
That is,
( e ( ϑ ) ) L
for all , ϑ A . A linearization for in (53) yields that
( e ( ϑ ) ) t + ( e ( ϑ ) t ) L
for all , ϑ , t A . Replacing by t in (54), we have
( ( e ( ϑ ) ) t [ e ( ϑ ) , t ] ) t + ( ( e ( ϑ ) t ) ) t [ e ( ϑ ) t , t ] L f o r a l l , ϑ , t A ,
which can be written as
( ( e ( ϑ ) ) t ) t ( [ e ( ϑ ) , t ] ) t + ( ( e ( ϑ ) t ) ) t [ e ( ϑ ) t , t ] L f o r a l l , ϑ , t A .
Using (54), we have
( [ e ( ϑ ) , t ] ) t [ e ( ϑ ) t , t ] L f o r a l l , ϑ , t A .
This impels that
( [ e ( ϑ ) , t ] t + [ e ( ϑ ) t , t ] ) [ , t ] [ e ( ϑ ) , t ] L f o r a l l , ϑ , t A .
Replacing by r in the last relation, we have
r ( [ e ( ϑ ) , t ] t + [ e ( ϑ ) t , t ] ) [ r , t ] [ e ( ϑ ) , t ] [ , t ] r [ e ( ϑ ) , t ] L f o r a l l , ϑ , t , r A .
Application of (55) gives that
[ , t ] r [ e ( ϑ ) , t ] L f o r a l l , ϑ , t , r A .
That is,
[ , t ] A [ e ( ϑ ) , t ] L f o r a l l , ϑ , t A .
The above relation is same as (46). Therefore using the same arguments as we have used after (46), we get e ( A ) L or A / L is a commutative integral domain. If e ( A ) L , then proof is done. On the other hand, if A / L is a commutative integral domain, then (54) reduces as
8 t e ( ϑ ) L f o r a l l , ϑ , t A .
Since c h a r ( A / L ) 2 , the above relation becomes
e ( ϑ ) A e ( ϑ ) L f o r a l l ϑ A .
The primeness of L forces that e ( A ) L . Thus the proof is completed now. □
The following results are immediate corollaries of Theorems 9 & 10.
 Corollary 12. 
Let A be a ring with involution * of the second kind and L a prime ideal of A such that S ( A ) Z ( A ) L . If A admits a derivation e such that [ [ e ( ) , ( ) * ] , ϑ ] L for all , ϑ A , then one of the following holds:
  • c h a r ( A / L ) = 2
  • e ( A ) L
  • A / L is a commutative integral domain.
 Corollary 13. 
[42, Theorem 3.7] Let A be a prime ring with involution * of the second kind such that c h a r ( A ) 2 . If A admits a derivation e such that [ e ( ) , ( ) * ] Z ( A ) for all A , then e = 0 or A is a commutative integral domain.
 Corollary 14. 
Let A be a prime ring with involution * of the second kind such that c h a r ( A ) 2 . If A admits a derivation e such that ( e ( ) ( ) * ) ( ) * = 0 for all A , then e = 0 .
 Theorem 12. 
Let A be a 2-torsion free semiprime ring with involution * of the second kind. If A admits a derivation e such that ( e ( ) ( ) * ) ( ) * = 0 for all A , then e = 0 .
 Proof. 
Given that
( e ( ) ( ) * ) ( ) * = 0 f o r a l l A .
By the semiprimeness of A , there exists a family L = { L α : α } of prime ideals such that α L α = ( 0 ) . For each L α in L , we have
( e ( ) ( ) * ) ( ) * L α f o r a l l A .
Application of Theorem 11 gives that e ( A ) L α . Thus, e ( A ) α L α = ( 0 ) and hence e = 0 . Thereby the proof is completed. □
We feel that Theorem 9 (resp. Theorem 11) can be proved without the assumption S ( A ) Z ( A ) L for any prime ideal L of an arbitray ring A , but unfortunately we are unable to do it. Hence, Theorem 9 leads the following conjecture.
Conjecture: Let A be a ring with involution * of the second kind and L a prime ideal of A . If A admits a derivation e such that [ [ e ( ) , ( ) * ] , ( ) * ] L for all A , then one of the following holds:
  • c h a r ( A / L ) = 2
  • e ( A ) L
  • A / L is a commutative integral domain.

5. A direction for further research

Throughout this section, we assume that k 1 , k 2 , m and n are fixed positive integers. Several papers in the literature evidence how the behaviour of some additive mappings is closely related to the structure of associative rings and algebras (cf.; [2,6,9,16,20,21][28,30,33] and [39]. A well-known result proved by Posner’s [46] states that a prime ring must be commutative if [ e ( ) , ] = 0 for all A , where e is a nonzero derivation of A . In [48,49], Vukman extended Posner’s theorem for commutators of order 2, 3 and described the structure of prings rings whose characteristic is not two and satisfying [ [ e ( ) , ] , ] = 0 for every A . The most famous and classical generalization of Posner’s and Vukman’s results are the following theorem due to Lanski [35] for k t h -commutators:
 Theorem 13. 
[35]Let m , n and k are fixed positive integers and A is prime ring. If a derivation e of A satisfies [ e ( m ) , m ] k = 0 for all I , where I is a nonzero left ideal of A , then e = 0 or A is commutative.
In [37], Lee and Shuie studied that if a noncommutative prime ring A admitting a derivation e such that [ e ( m ) n , r ] k = 0 for all I , where I is a non zero left ideal, then e = 0 except when A M 2 ( G F ( 2 ) ) . In the year 2000, Carini and De Filippis [23] studied Posner’s classical result for power central values. In particular, they discussed this situation for A of characteristic not two and proved that if ( [ e ( ) , ] ) n Z ( A ) for all L , a noncentral Lie ideal of A , then A satisfies s 4 . In 2006, Wang and You [51] mentioned that the restriction of characteristic need not necessary in Theorem 1.1 of [23]. More precisely, they proved the following result.
 Theorem 14. 
Let A be a noncommutative prime ring and L be a noncentral Lie ideal of A . If A admits a derivation e satisfies ( [ e ( m ) , m ] ) n Z ( A ) for all L , then A satisfies s 4 , the standard identity in 4 variables.
Motivated by these two results, Wang [50] studied the similar condition for A of characteristic not two and obtained the same conclusion. In fact, he proved the following results.
 Theorem 15. 
Let A be a noncommutative prime ring of characteristic not two. If A admits a nonzero derivation e satisfies ( [ e ( m ) , m ] n ) k Z ( A ) for all A , then A satisfies s 4 , the standard identity in 4 variables.
In our main results (Theorems 1, 3, 5, 9 and 10), we investigate the structure of the qutiont rings A / L , where A is an arbitrary ring and L is a prime ideal of A . Nevertheless, there are various interesting open problems related to our work. In this final section, we will propose a direction for future further research. In view of the above mentioned results and our main theorems, the following problems remains unanswered.
 Problem 1. 
Let A be a ring of suitable characteristic with involution * of the second kind and L a prime ideal of A such that S ( A ) Z ( A ) L . Next, let f : A A be a mapping satisfying [ f ( ) , ( ( ) * ) m ] n Z ( A ) or L for all A . Then, what we can say about the structure of A and f ?
 Problem 2. 
Let A be a ring of suitable characteristic with involution * of the second kind and L a prime ideal of A such that S ( A ) Z ( A ) L . Next, let e : A A be a derivation satisfying [ e ( ) , ( ) * ] n Z ( A ) or L for all A . Then, what we can say about the structure of A and e ?
 Problem 3. 
Let A be a ringof suitable characteristic with involution * of the second kind and L a prime ideal of A such that S ( A ) Z ( A ) L . Next, let e : A A be a derivation satisfying ( [ e ( x k 1 ) , ( ( ) * ) k 2 ] n ) m Z ( A ) or L for all A . Then, what we can say about the structure of A and e ?
 Problem 4. 
Let A be a ring of suitable characteristic with involution * of the second kind and L a prime ideal of A such that S ( A ) Z ( A ) L . Next, let e : A A be a derivation satisfying ( e ( k 1 ) n ( ( ) * ) ) k 2 ) m Z ( A ) or L for all A . Then, what we can say about the structure of A and e ?

Author Contributions

All authors made equal contributions.

Data Availability Statement

Data sharing is not applicable as no datasets were generated or analyzed during the current study.

Acknowledgments

This work was supported by TUBITAK, the Scientic and Technological Research Council of Turkey, under the program 2221-Fellowship for Visiting Professor/Scientists at Karamanoglu Mehmetbey University (KMU), Turkey. So, the second and third authors, gratefully acknowledge the TUBITAK for technical and financial support.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

References

  1. Abdelkarim, B. and Ashraf, M., Identities related to generalized derivations in prime *-rings, Georgian Math. J. 28(2) (2021), 193–205.
  2. Alahmadi, A., Alhazmi, H., Ali, S., Dar, A. N. and Khan, A. N., Additive maps on prime and semiprime rings with involution, Hacet. J. Math. Stat. 49(3) (2020), 1126–1133. [CrossRef]
  3. Albaş, E., Argaç, N. and De Fillipis, V., Generalized derivations with Engel conditions on one sided ideals, Comm. Algebra 36(6) (2008), 2063–2071.
  4. Ali, A., Rehman, N. and Ali, S., On Lie ideals with derivations as homomorphisms and anti-homomorphisms, Acta Math. Hungar. 101(1-2) (2003), 79–82.
  5. Almahdi, F. A. A., Mamouni, A. and Tamekkante, M., A generalization of Posner’s theorem on derivations in rings Indian J. Pure Appl. Math. 51(1) (2020), 187–194.
  6. Ali, S. and Dar, N. A., On *-centralizing mappings in rings with involution, Georgian Math. J. 21(1) (2014), 25–28. [CrossRef]
  7. Ali, S., Dar, N. A. and Asci, M., On derivations and commutativity of prime rings with involution, Georgian Math. J. 23(1) (2016), 9–14. [CrossRef]
  8. Ali, S., Dar, N. A., Dušan, P.,On Jordan *-mappings in rings with involution, Journal of the Egyptian Mathematical Society 24 (2016), 15–19.
  9. Ali, S., Koam, A. N. A. and Ansari, M. A., On *-differential identities in prime rings with involution, Hacet. J. Math. Stat. 49(2) (2020), 708–715. [CrossRef]
  10. Ali, S., Alhazmi, H. Dar, N. A. and Khan, A. N., A characterization of additive mappings in rings with involution, De Gruter Proc. Math. Berlin (2018), 11–24.
  11. Anderson, F. W., Lecturs on noncommutative rings, Univ. of Oregon, 2002.
  12. Argac, N., On prime and semiprime rings with derivations, Algebra, Colloq., 13(3) (2006), 371–380. [CrossRef]
  13. Ashraf, M. and Siddeeque, M. A., Posner’s first theorem for *-ideals in prime rings with involution, Kyungpook Math. J. 56(2) (2016), 343–346.
  14. Ashraf, M., Ali, S., Haetinger, C., On derivations in rings and their applications, The Aligarh Bull. Math. 25 (2006),79–-107.
  15. Bell, H. E. and Daif, M. N., On derivations and commutativity in prime rings, Acta Math. Hungar. 66(4) (1995), 337–343. [CrossRef]
  16. Bell, H. E. and Daif, M. N., On commutativity and strong commutativity-preserving maps, Canad. Math. Bull. 37(4) (1994), 443–447. [CrossRef]
  17. Bell, H. E. and Daif, M. N., Remarks on derivations on semiprime rings, Internat. J. Math. Math. Sci. 15(1) (1992), 205–206. [CrossRef]
  18. Bell, H. E. and Kappe, L. C., Rings in which derivations satisfy certain algebraic conditions, Acta Math. Hungar. 53(3-4) (1989), 339–346. [CrossRef]
  19. Bell, H. E. and Martindale III, W. S., Centralizing mappings of semiprime rings, Canad. Math. Bull. 30(1) (1987), 92–101.
  20. Brešar, M., Commuting traces of biadditive mappings, commutativity-preserving mappings and Lie mappings, Trans. Amer. Math. Soc. 335(2) (1993), 525–546.
  21. Bresˇar, M., Centralizing mappings and derivations in prime rings, J. Algebra, 156 (1993), 385–394. [CrossRef]
  22. Brešar, M. and Miers, C. R., Strong commutativity preserving maps of semiprime rings, Canad. Math. Bull. 37(4) (1994), 457–460. [CrossRef]
  23. Carini, L. and De Fillipis, V., Commutators with power central values on Lie ideals, Pacific. J. Math. 193 (2006), 269–273.
  24. Creedon, T., Derivations and prime ideals, Math. Proc. R. Ir. Acad. 98A(2) (1998), 223–225.
  25. Dar, N. A. and Ali, S., On *-commuting mappings in rings with involution, Turk. J. Math. 40 (2016), 884–894.
  26. De Filippis, V., Generalized derivations with Engel condition on multilinear polynomials, Israel J. Math. 171 (2009), 325–348.
  27. De Filippis, V., Left annihilator of commutators with derivations on right ideals, Comm. Algebra 31(10) 2003, 5003-5010.
  28. Deng, Q. and Ashraf, M., On strong commutativity preserving mappings, Results Math. 30(3-4) (1996), 259–263. [CrossRef]
  29. Deng, Q. and Bell, H. E., On derivations and commutativity in semiprime rings, Comm. Algebra 23 (1995), 3705-3713. [CrossRef]
  30. Dhara, B. and Ali, S. On n-centralzing generalized derivations in semiprime rings with applications to C*-algebras, J. Algebra Appl. 6 (2012), 11pages.
  31. Divinsky, N. J., On commuting automorphisms of rings, Trans. Roy. Soc. Canada Sect. III 49 (1955), 19–22.
  32. Herstein, I. N., Ring with involution, University of Chicago Press, Chicago, 1976.
  33. Lanski, C., Differential identities, Lie ideals, and Posner’s theorems, Pacific J. Math. 134(2) (1988), 275–297.
  34. Lanski, C., An Engel condition with derivation, Proc. Amer. Math. Soc. 118 (1993), 731–734.
  35. Lanski, C., An Engel condition with derivation for left ideals, Proc. Amer. Math. Soc. 125 (1997), 339–345. [CrossRef]
  36. Lee, T. K. and Lee, T. C., Commuting additive mappings in semiprime rings, Bull. Inst. Math. Acad. Sinica 24 (1996), 259–268.
  37. Lee, T. K. and Shiue, W. K., A result on derivations with Engel condition in prime rings, Southeast Asian Bull. Math. 23(3) (1999), 437–446.
  38. Luh, J., A note on commuting automorphisms of rings, Amer. Math. Monthly 77 (1970), 61–62.
  39. Mamouni, A., Nejjar, B. and Oukhtite, L., Differential identities on prime rings with involution, J. Algebra Appl. 17(9) (2018), 1850163, 11pp. [CrossRef]
  40. Mamouni, A., Oukhtite, L. and Zerra, M., On derivations involving prime ideals and commutativity in rings, São Paulo J. Math. Sci. 14(2) (2020), 675–688. [CrossRef]
  41. Mir, H. E., Mamouni, A. and Oukhtite, L., Commutativity with algebraic identities involving prime ideals, Commun. Korean Math. Soc. 35(3) (2020), 723–731.
  42. Nejjar, B., Kacha, A., Mamouni, A. and Oukhtite, L., Commutativity theorems in rings with involution, Comm. Algebra 45(2) (2017), 698–708. [CrossRef]
  43. Mayne, J., Centralizing automorphisms of Lie ideals in prime rings, Canad. Math. Bull. 35 (1992), 510–-514. [CrossRef]
  44. Mayne, J., Centralizing automorphisms of prime rings, Canad. Math. Bull. 19 (1976), 113–117. [CrossRef]
  45. Oukhtite, L., Posner’s second theorem for Jordan ideals in ring with involution, Expo. Math. 4(29) (2011), 415–419. [CrossRef]
  46. Posner, E. C., Derivations in prime rings, Proc. Amer. Math. Soc. 8(6) (1957), 1093–1100.
  47. Sˇemrl, P., On Jordan *-derivations and application, Colloq. Math., 59 (1990), 241–251. [CrossRef]
  48. Vukman, J., Commuting and centralizing mappings in prime rings, Proc. Amer. Math. Soc. 109 (1990), 47–-52.
  49. Vukman, J. and Ashraf M., Derivations and commutativity of prime rings, Aligarh Bull. Math 18 (1999), 24–28.
  50. Wang, Y., A generalization of Engel conditions with derivations in rings, Comm. Algebra 39 (2011), 2690–2696. [CrossRef]
  51. Wang, Y. and You, H., A note on commutators with power central values on Lie ideals in rings, Acta. Math. Sinica. English Series 22 (2006), 1715–1720.
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.
Copyright: This open access article is published under a Creative Commons CC BY 4.0 license, which permit the free download, distribution, and reuse, provided that the author and preprint are cited in any reuse.
Prerpints.org logo

Preprints.org is a free preprint server supported by MDPI in Basel, Switzerland.

Subscribe

© 2024 MDPI (Basel, Switzerland) unless otherwise stated