# Top PDF Jordan k Derivations on Lie Ideals of Prime Γ Rings

### Jordan k Derivations on Lie Ideals of Prime Γ Rings

Lemma 2.3 . Let M be a 2- torsion free Γ- ring satisfying the condition (*) and let U be a Lie ideal of M such that uαu ∈ U for all u ∈ U and α ∈ Γ. Let d : M → M be a Jordan k-derivation on U of M . Then for all u, v, w ∈ U and α, β ∈ Γ , we have the following : (i)d(uαv +vαu) = d(u)αv +uk(α)v +uαd(v) +d(v)αu + vk(α)u + vαd(u)

### Lie Ideals on Prime Γ-Rings with Jordan Right Derivations

Abstract. In this paper, we analyzed the basic properties and related theorem of Lie ideals on prime Γ -rings with Jordan right derivations. We mainly focused on the characterizations of 2-torsion free prime Γ -rings by using Lie ideals and Jordan right derivations. Our main objective is to prove the theorem that if ܯ be a 2 -torsion free prime Γ -ring and ܷ be a Lie ideal of ܯ such that ݑߙݑ ∈ ܷ for all ݑ, ݒ ∈ ܷ and ߙ ∈ Γ and ݀: ܯ → ܯ is an additive mapping such that ݀(ݑߙݑ) = 2 ݀(ݑ)ߙݑ for all ݑ ∈ ܷ and ߙ ∈ Γ , then ݀(ݑߙݒ) = ݀(ݑ)ߙݒ + ݀(ݒ)ߙݑ for all ݑ, ݒ ∈ ܷ and ߙ ∈ Γ .

### On Jordan ideals and left (θ,θ) derivations in prime rings

1. Introduction. Throughout the present paper, R will denote an associative ring with centre Z(R). We will write for all x, y ∈ R, [x, y] = xy − yx and x ◦ y = xy + yx for the Lie product and Jordan product, respectively. A ring R is said to be prime if aRb = (0) implies that a = 0 or b = 0. A ring R is said to be 2-torsion-free if whenever 2a = 0, with a ∈ R, then a = 0. An additive subgroup J of R is said to be a Jordan ideal of R if u ◦ r ∈ J, for all u ∈ J and r ∈ R. An additive mapping d : R → R is called a derivation (resp., Jordan derivation) if d(xy) = d(x)y + xd(y) (resp., d(x 2 ) = d(x)x + xd(x)) holds for all x, y ∈ R. Let θ, φ be endomorphisms of

### Derivations Acting as Homomorphisms and as Anti homomorphisms in σ Lie Ideals of σ Prime Gamma Rings

In [4], Bell and Kappe proved that if d is a derivation of a semiprime ring R which is either an endomorphism or an anti-endomorphism on R, then d = 0; whereas, the behavior of d is somewhat restricted in case of prime rings in the way that if d is a derivation of a prime ring R acting as a homomorphism or an anti-homomorphism on a non-zero right ideal U of R, then d = 0. Asma et. al. [1] extended this result of prime rings on square closed Lie ideals. Afterwards, the said result was extended to σ-prime rings by Oukhtite et. al. in [11].

### Lie Ideals and Generalized Derivations in Semiprime Rings

A well known result of Posner [18] states that a prime ring admitting a nonzero centralizing derivation must be commutative. This theorem indicates that the global structure of a ring R is often tightly connected to the behaviour of additive mappings defined on R. Following this line of investigation, several authors studied derivations and generalized derivations acting on appropriate subsets of the ring.

### Generalized derivations in prime rings

Firstly, E.C. Posner [17] proved pioneer results on derivation in prime rings. He established relation between derivation on a ring and structure of that ring. Many authors have generalized Posner’s theorems, for suitable subsets of R, as ideal, left ideal, Lie ideal and Jordan ideal, further information can be found in ([2], [5],[9],[12],[15],[16], [18],[20]). In [6] Bell and Kappe proved that if a deriva- tion d of a prime ring R which acts as homomorphisms or anti-homomorphisms on a nonzero right ideal of R then d = 0 on R. In [14] Nadeem-ur rehman generalized Bell and Kappe result by taking generalized derivation instead of derivation. Precisely he proved, let R be a 2-torsion free prime ring and I be a non zero ideal of R. Suppose F : R → R is a nonzero generalized derivation with non zero derivation d. If F acts as a homomorphism or anti-homomorphism on I then R is commutative. In this sequence, in 2001 Ashraf and Rehman [4], proved that if R is a prime ring with a non-zero ideal I of R and d is a derivation of R such that either d(xy) ±xy ∈ Z(R) for all x, y ∈ I or d(xy)±yx ∈ Z(R) for all x, y ∈ I, then R is commutative. Again, Asraf et al. [3] proved that if R is a prime ring which is 2 torsion free and F is a generalized derivation associated with derivation d on R. If F satisfies any one of the following conditions: (i) F(xy) − xy ∈ Z(R); (ii) F (xy) − yx ∈ Z(R); (iii) F (x)F (y) − xy ∈ Z(R); (iv) F(x)F(y) − yx ∈ Z(R), for all x, y ∈ I, where I is an ideal of R, then R is commutative. Recently, Albas [17] studied following identities in prime rings: (i) F(xy) ± F (x)F (y) ∈ Z(R); (ii) F (xy) ± F(y)F (x) ∈ Z (R), for all x, y ∈ I, a non zero ideal of R.

### Generalized Derivations of Prime Rings

In [3], Bergen et al. investigate the relationship between the derivations and Lie ideals of a prime ring, and obtain some useful results. In [4], P. H. Lee and T. K. Lee get six suﬃcient conditions of central Lie ideal, which extend some results of commutativity on a prime ring. Motivated by the above, in this paper, we extend M. Ashraf ’s results to a Lie ideal of a prime ring. Throughout this paper, R will be a prime ring and U will always denote a Lie ideal of R.

### Left Generalized Derivations and Commutativity of Prime Rings

Where is an inner derivation. An additive mapping : → is called a generalized derivation associated with a derivation if ( ) = ( ) + ( ) , for all , ∈ . An additive mapping : → is called a left generalized derivation associated with a derivation if ( ) = ( ) + ( ) , for all , ∈ . It should be interesting to extend some results concerning these notions to left generalized derivations. Throughout the present paper we shall make extensive use of the following basic commutator identities without any specific mention:

### Skew Commuting Derivations of Noncommutative Prime Rings

Derivations on rings help us to understand rings better and also derivations on rings can tell us about the structure of the rings. For instance a ring is commutative if and only if the only inner derivation on the ring is zero. Also derivations can be helpful for relating a ring with the set of matrices with entries in the ring (see, [5]). Derivations play a significant role in determining whether a ring is commutative, see ( [1],[3],[4],[18],[19] and [20]).Derivations can also be useful in other fields. For example, derivations play a role in the calculation of the eigenvalues of matrices (see, [2]) which is important in mathematics and other sciences, business and engineering. Derivations also are used in quantum physics(see, [18]). Derivations can be added and subtracted and we still get a derivation, but when we compose a derivation with itself we do not necessarily get a derivation. The history of commuting and centralizing mappings goes back to (1955) when Divinsky [6] proved that a simple Artinian ring is commutative if it has a commuting nontrivial automorphism. Two years later, Posner[7]has proved that the existence of a non- zero centralizing derivation on prime ring forces the ring to be commutative (Posner's second theorem).Luch [8]generalized the Divinsky result, we have just mentioned above, to arbitrary prime ring. In[9] M.N.Daif, proved that, let R be a semiprime ring and d a derivation of R with d 3 ≠0 .If [d

### Γ-Semigroups in which Prime Γ-Ideals are Maximal

Γ- semigroup was introduced by Sen and Saha [10] as a generalization of semigroup. Anjaneyulu. A [1], [2] and [3] initiated the study of pseudo symmetric ideals, radicals and semi pseudo symmetric ideals in semigroups. Giri and Wazalwar [4] intiated the study of prime radicals in semigroups. Madhusudhana Rao, Anjaneyulu and Gangadhara Rao [5], [6], [7] and [8] initiated the study of prime radicals and semi pseudo symmetric Γ-ideals in Γ-semigroups, primary and semiprimary Γ-ideals and pseudo symmetric Γ-ideals in Γ-semigroups. In this paper we characterize Quasi Commutative Γ-semigroup, semi pseudo symmetric Γ-semigroups and quasi commutative Γ-semigroups containing cancellable elements in which proper prime Γ-ideals are maximal. We first cite a wide class of primary Γ-semigroups.

### Semi-Prime Ideals of Gamma Rings

Although it is possible to prove results about semi-prime ideals that are analogous to all of those established for prime ideals proceeding section, we shall present only a few that are essential for later applications. First, let us state the following theorem whose proof we omit since it can be established by very easy modifications of the proof of theorem 3.2.

### On Prime and Semiprime Rings with Additive Mappings and Derivations

such that [d(x), x] ∈ Z(R) for all x ∈ R, then R is commutative. An analogous result for centralizing automorphisms on prime rings was obtained by Mayne [28]. A number of authors have extended these theorems of Posner and Mayne; they have showed that derivations, automorphisms, and some related maps cannot be centralizing on certain subsets of noncommutative prime (and some other) rings. For these results we refer the reader to ([10], [12], [15], [25], where further references can be found). In [14], the description of all centralizing additive maps of a prime ring R of characteristic not 2 was given and subsequently in [4] the characterization for semiprime rings of characteristic not 2 was given. It was shown that every such map f is of the form f (x) = λx + µ(x), where λ ∈ C, the extended centroid of R, and µ is an additive map of R into C (see also [12] where similar results for some other rings are presented). Recently, some authors have obtained commutativity of prime and semiprime rings with derivations and generalized derivations satisfying certain polynomial identities (viz., [1], [2], [3], [5], [6], [7], [8], [9], [16], [17], [18], [19], [20], [24], [25], [27], [29], [31] and [32]). The main objective of the present paper is to investigate commutativity of prime and semiprime rings satisfying certain identities involving additive mappings and derivations.

### Products of derivations which act as Lie derivations on commutators of right ideals

This note is motived by the previous cited results. Our main theorem gives a general- ization of Lanski’s result to the case when (dδ) is a Lie derivation of the subset [I,I ] into R, where I is a nonzero right ideal of R and the characteristic of R is di ﬀ erent from 2. The statement of our result is the following.

### On Prime Gamma Near Rings with Generalized Derivations

Γ-near-ring with multiplicative centre ZN. Recall that a Γ-near-ring N is prime if xΓNΓy 0 implies x 0 or y 0. An additive mapping d : N → N is said to be a derivation on N if dxαy xαdy dxαy for all x, y ∈ N, α ∈ Γ, or equivalently, as noted in 1, that dxαy dxαy xαdy for all x, y ∈ N, α ∈ Γ. Further, an element x ∈ N for which dx 0 is called a constant. For x, y ∈ N, α ∈ Γ, the symbol x, y α will denote the commutator xαy − yαx, while the symbol x, y will denote the additive-group commutator x y − x − y. An additive mapping f : N → N is called a generalized derivation if there exits a derivation d of N such that fxαy f xαy xαdy for all x, y ∈ N, α ∈ Γ. The concept of generalized derivation covers also the concept of a derivation.

### Projective prime ideals and localisation in pi rings

One of the results required along the way is of independent interest. We prove (Lemma 3.1) that in a Noetherian ring an ideal with zero right annihilator and satisfying (a weak form of ) the right AR-property contains a right regular element. Our methods require Theorem A to be first proved for a maximal ideal. Extending the result to a general prime ideal presents a technical challenge. Since it is not yet known whether the cliques in a Noetherian PI-ring are localisable, a direct localisation approach is not available to us. We sidestep this difficulty by employing a trick of Goodearl and Stafford. This device guarantees that the prime ideal being examined extends to a prime ideal which belongs to a localisable clique in a polynomial extension of the given Noetherian PI-ring. With the authors’ permission an account of this method is included here.

### On (?,?)-Derivations and Commutativity of Prime and Semi prime ?-rings Afrah Mohammad Ibraheem

The set Z(R) = {a ∈R| aαb = bαa, ∀b ∈R, andα∈Γ} is called the center of R.A Γ-ring R is called prime if aΓRΓb = 0 with a, b ∈R implies a = 0 or b = 0, and R is called semi prime if aΓRΓa = 0 with a ∈R implies a = 0. The notion of a (resp. semi-) prime Γ-ring is an extension for the notion of a (resp. semi-) prime ring. In [1] F.J.Jing defined a derivation on Γ-ring as follows, an additive map d from a Γ- ring R into itself is called a derivation on R if d(aαb) = d(a)αb +aαd(b) , holds for all a,b∈R and α∈Γ, and in [2] S.

### Symmetric Skew Reverse n-Derivations on Prime Rings and Semiprime rings

Theorem.3.3. Let n ≥2 be any fixed positive integer. Let R be an n! – torsion free prime ring and I be any non zero two sided ideal of R. Suppose that there exist a non zero symmetric skew reverse n-derivation Ä: R n →R, associated with an antiautomorphism á * . Let ä denote the trace of  such that ä is commuting on I and [ ä (x), á * (x)]  Z(R), for all x  I then R must be commutative.

### (1,α)- Derivations in Prime г - near Rings

The notion of a ring a concept more general than a ring was definedby Nobusawa [7]. As a generalization of near rings, rings were introduced by satyanarayana[8]. The derivation of a near ring has been introduced by Bell and Mason[2]. They obtained some basic properties of derivations in near ring. The recent literature contains numerous results on commutativity in prime and semiprime rings admitting suitably constrained derivations and generalized derivations, and several authors have proved comparable results on near-rings. Some of our results, which deal with conditions on derivations, extend earlier commutativity results involving similar conditions on derivations and semi-derivations.