Characterization of 3-prime near-rings via multiplicative derivations

The main purpose of this paper is to show that zero symmetric prime left near-rings satisfying certain identities are commutative rings . As a consequence of the results obtained ,we prove several commutativity theorems about generalized n-derivatios for prime near-rings.

In this paper, we introduce the notion of α-n derivations in prime nearrings, In [3]Samman. M.S interpreted α-derivation of A and studied some properties of this derivation. In [1], M.Ashraf and M.A.Siddeeque interpreted n-derivation in nearing and studied some properties included there, influenced by this concept. α-n-derivation of nearing A., which gives a generalization of nderivation of nearing is defined in this study. The main purpose is to show that a prime nearing A with some identities on α-n-derivations satisfies some important properties. Our results are generalized to many previously results on prime nearing with derivations, n-derivations and α-derivations, initially we begin with necessary lemmas which are essential for developing the proofs our main results. In[3] Samman M.S.proved that if d is an α – derivation of a prime nearringA in a way that d replace with α, then d2 = 0 implies d = 0 and The composition of α – derivations obtained, We have extended this result in the setting of an α - nderivation in near rings.In this paper, we introduce the notion of α-n derivations in prime nearrings, In [3]Samman. M.S interpreted α-derivation of A and studied some properties of this derivation. In [1], M.Ashraf and M.A.Siddeeque interpreted n-derivation in nearing and studied some properties included there, influenced by this concept. α-n-derivation of nearing A., which gives a generalization of nderivation of nearing is defined in this study. The main purpose is to show that a prime nearing A with some identities on α-n-derivations satisfies some important properties. Our results are generalized to many previously results on prime nearing with derivations, n-derivations and α-derivations, initially we begin with necessary lemmas which are essential for developing the proofs our main results. In[3] Samman M.S.proved that if d is an α – derivation of a prime nearringA in a way that d replace with α, then d2 = 0 implies d = 0 and The composition of α – derivations obtained, We have extended this result in the setting of an α -...

This paper investigates the analysis of prime near-ring by explore the detailed structure of the generalized derivations that satisfy some specific assumptions. Let N be a prime near-rings and G be a generalized reverse derivative associated with mapping d on N. An additive mapping d:N→N is said to be a derivation on N if d(xy)=d(x)y+xd(y) for all x,y ∈N. A mapping G: N→N associated with derivation d is called a generalized derivation on N if G(xy)=G(x)y+xd(y) for all x,y∈N. Also, a mapping d: N→N is said to be a reverse derivation on N if d(xy)=d(y)x+yd(x) for all x,y ∈N and a mapping G:N→N associated with reverse derivation d is said to be a generalized reverse derivation on N if G(xy)=G(y)x+yd(x) for all x,y∈N. We prove some results on commutativity of prime near-rings involving generalized reverse derivations. In addition, we prove that; for prime near-rings N, if d(x)d(y)±xy=0 for all x,y∈N then d=0 where d is a skew- derivation associated with an automorphism β∶ N→N. Furthermore, for a prime near-ring N with generalized derivative G associated with mapping d on N, if G(x)G(y)±xy=0 for all x,y∈G then d=0.

Let N be a 3-prime left near-ring with multiplicative center Z, f be a generalized (σ,τ)- derivation on N with associated (σ,τ)-derivation d and I be a semigroup ideal of N. We proved that N must be a commutative ring if f(I)⊂Z or f act as a homomorphism or f act as an anti-homomorphism.

On derivations and commutativity in prime near-rings

Some results by Bell and Mason on commutativity in near-rings are generalized. Let N be a prime right near-ring with multiplicative center Z and let D be a (σ,τ)-derivation on N such that σD = Dσ and τD = Dτ. The following results are proved: (i) If D(N) ⊂ Z or [D(N), D(N)] = 0 or [D(N), D(N)]σ,τ = 0 then (N, +) is abelian; (ii) If D(xy) = D(x)D(y) or D(xy) = D(y)D(x) for all x, y ∈ N then D = 0.

Let N be a left near-ring and S be a nonempty subset of N. A mapping F from N to N is called commuting on S if [F(x),x] = 0 for all x € S. The mapping F is called strong commutativity preserving (SCP) on S if [F(x),F(y)] = [x,y] for all x, y € S. In the present paper, firstly we generalize the well known result of Posner which is commuting derivations on prime rings to generalized derivations of semiprime near-rings. Secondly, we investigate SCP-generalized derivations of prime near-rings.

The analog of Posner's theorem on the composition of two derivations in prime rings is proved for 3-prime near-rings. It is shown that if is a nonzero derivation of a 2-torsionfree 3-prime near-ring N and an element a ∊ N is such that axd = xda for all x ∊ N, then a is a central element. As a consequence it is shown that if d and d2 are nonzero derivations of a 2-torsionfree 3-prime near-ring N and xd1yd2 = yd2xd1 for all x, y ∊ N, then N is a commutative ring. Thus two theorems of Herstein are generalized

Near-rings are generalize from rings. A research on near-ring is continous included a research on prime near-rings and one of this research is about derivation on prime near-rings. This article will reviewing about relation between derivation on prime near-rings and commutativity in rings with literature review method. The result is prime near-rings are commutative rings if a nonzero derivation d on N hold one of this following conditions: (i) , (ii) , (iii) , (iv) , (v) , (vi) , for all , with is non zero semigroup ideal from .

The purpose of this paper is to investigate two sided $\alpha $-derivations satisfying certain differential identities on 3-prime near-rings. Some well-known results characterizing commutativity of 3-prime near-rings by derivations (semi-derivations) have been generalized. Furthermore, examples proving the necessity of the 3-primeness hypothesis are given.

There is a large body of evidence showing that the existence of a suitably-constrained derivation on a 3-prime near-ring forces the near-ring to be a commutative ring. The purpose of this paper is to study generalized semiderivations which satisfy certain identities on 3-prime near-ring and generalize some results due to [H. E. Bell and G. Mason, On derivations in near-rings, North-Holland Math. Stud. 137 (1987) 31–35; H. E. Bell, On prime near-rings with generalized derivation, Int. J. Math. Math. Sci. 2008 (2008), Article ID: 490316, 5[Formula: see text]pp; A. Boua and L. Oukhtite, Some conditions under which near-rings are rings, Southeast Asian Bull. Math. 37 (2013) 325–331]. Moreover, an example is given to prove that the necessity of the 3-primeness hypothesis imposed on the various theorems cannot be marginalized.

Let be a 3-prime near-ring and are endomorphisms. In this paper, we generalize a few results concerning generalized derivations and two sided α -generalized derivations of 3-prime near rings to generalized (α, β) -derivations. Cases demonstrating the need of the 3-primeness speculation is given. When (resp. ), one can easily obtain the main results of [1] (resp. [4]).

We introduce the notion of a strongly prime near-ring module and then characterize strongly prime near-rings in terms of strongly prime modules. Furthermore, we define a \({\mathcal{T}}\)-special class of near-ring modules and then show that the class of strongly prime modules forms a \({\mathcal{T}}\)-special class. \({\mathcal{T}}\)-special classes of strongly prime modules are then used to describe the strongly prime radical.

In the present paper, we introduce the notion of generalized ( σ , τ ) {(\sigma,\tau)} -n-derivations in a near-ring N and investigate a property involving generalized ( σ , τ ) {(\sigma,\tau)} -n-derivations of a prime near-ring N, which makes N a commutative ring. Additive commutativity of a prime near-ring N satisfying certain identities involving generalized ( σ , τ ) {(\sigma,\tau)} -n-derivations is also obtained.

The concept of differential algebra has been initiated before many years ago. This research topic has inspired a lot of authors to its study with different algebraic structures such as rings or semi-rings. Their studies provided many good results in this field some of which by extending from previous works and others by introducing new notion in this direction. In this article a type of rings are called prime near-rings have been considered. In particular, we introduced the idea of generalized (theta,theta)-semi-derivation of prime near-rings and studied the commutativity of such types of rings by using this notion.