Prime Near-rings Research Articles

On derivations and commutativity in prime near-rings

In the present paper it is shown that zero symmetric prime right near-rings satisfying certain identities are commutative rings.

Commutativity of rings and near-rings with generalized derivations

In this paper we prove some theorems of commutativity for prime rings or 3-prime near-rings with a suitably constrained generalized derivation. As a consequence of the results obtained, we prove several commutativity theorems for 3-prime near-rings and prime rings. Also, we prove some other results about the generalized derivation either on a near-ring or on a ring.

$(\theta,\phi)$-derivations as homomorphisms or as anti-homomorphisms on a near ring

Let $N$ be a near ring. An additive mapping $d:N\longrightarrow N$ is said to be a $(\theta,\phi)$-derivation on $N$ if there exist mappings $\theta,\phi:N\longrightarrow N$ such that$d(xy)=\theta(x)d(y)+d(x)\phi(y)$ holds for all $x,y \in N$. In the context of 3-prime and 3-semiprime nearrings, we show that for suitably-restricted $\theta$ and $\phi$, there exist no nonzero $(\theta,\phi)$-derivations which act as a homomorphism or an anti-homomorphism on $N$ or a nonzero semigroup ideal of $N$.

Commutativity of near-rings with derivations by using algebraic substructures

In this paper we use subsets and algebraic substructures of a 3-prime nearring R admitting a derivation d to study the commutativity of the subsets, the algebraic substructures and the near-ring R under suitable conditions on d, R and the algebraic substructures. The results obtained in this paper generalize several commutativity theorems due to some authors.

Some Conditions under Which Prime near-Rings Are Commutative Rings

Let N be a prime left near-ring, and I be a nonzero semigroup ideal of N . We prove that if N admits a derivation d satisfying any one of the following properties: (i)d([x;y]) = [x;y], (ii)d([x;y]) = [d(x);y], (iii) [d(x);y] = [x;y], (iv)d(x y) = x y, (v)d(x) y = x y and (vi)d(x) y = x y for all x;y2 I; then N is a commutative ring. Moreover, example proving the necessity of the primeness condition is given.

Semigroup ideal in Prime Near-Rings with Derivations

In this paper we generalize some of the results due to Bell and Mason on a near-ring N admitting a derivation D , and we will show that the body of evidence on prime near-rings with derivations have the behavior of the ring. Our purpose in this work is to explore further this ring like behavior. Also, we show that under appropriate additional hypothesis a near-ring must be a commutative ring.

Semigroup ideal in Prime Near-Rings with Derivations

In this paper we generalize some of the results due to Bell and Mason on a near-ring N admitting a derivation D , and we will show that the body of evidence on prime near-rings with derivations have the behavior of the ring. Our purpose in this work is to explore further this ring like behavior. Also, we show that under appropriate additional hypothesis a near-ring must be a commutative ring.

Strongly Prime Near-Ring Modules

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.

Notes on the commutativity of prime near-rings

Notes on the commutativity of prime near-rings

Different Prime R-Ideals

It is well known that there are several non-equivalent types of prime near-rings which are all equivalent in the case of associative rings. In this paper we introduce various characterizations of prime modules in a zero-symmetric near-ring R. The connection of a prime R-ideal P of a module M and the ideal (P:M) of the near-ring R is also investigated.

On Prime and Semiprime Near-Rings with Generalized Derivation

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.

On Prime Near-Rings with Generalized Derivation

LetNbe a 3-prime 2-torsion-free zero-symmetric left near-ring with multiplicative centerZ. We prove that ifNadmits a nonzero generalized derivationfsuch thatf(N)⊆Z, thenNis a commutative ring. We also discuss some related properties.

On Zero-Symmetric Left Centrally Prime Near-Rings

On Zero-Symmetric Left Centrally Prime Near-Rings

Different nil radicals for near-rings

It is well known that there are a number of different nonequivalent definitions of a prime near-ring. In this paper we show that this give rise to a number of different upper nil radicals which all coincide with the usual upper nil radical in the case of associative rings.

Some properties of prime near-rings with (σ,τ)-derivation

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.

Near-Ring Radicals and Class Pairs

For near-ring ideal mappings ρ1and ρ2, we investigate radical theoretical properties of and the relationship among the class pairs (ρ1: ρ2), [Formula: see text] and (ℛρ2: ℛρ1). Conditions on ρ1and ρ2are given for a general class pair to form a radical class of various types. These types include the Plotkin and KA-radical varieties. A number of examples are shown to motivate the suitability of the theory of Hoehnke-radicals over KA-radicals when radical pairs of near-rings are studied. In particular, it is shown that [Formula: see text] forms a KA-radical class, where [Formula: see text] denotes the class of completely prime near-rings and [Formula: see text] the class of 3-prime near-rings. This gives another near-ring generalization of the 2-primal ring concept. The theory of radical pairs are also used to show that in general the class of 3-semiprime near-rings is not the semisimple class of the 3-prime radical.

Maximal Right Nearring of Quotients and Semigroup Generalized Polynomial Identity

In the present paper, we study the maximal right nearring of quotients of 3-semiprime nearrings. We establish its existence by an explicit construction and give an axiomatic description. We show that the operation of taking the maximal right nearring of quotients is idempotent. Further, we show that the maximal right nearring of quotients of a 3-prime nearring N with a minimal left N-subgroup is a centralizer nearring determined by some fixed point free group of automorphisms. Finally, we describe equiprime nearrings with nontrivial multilinear semigroup generalized polynomial identities. They are the equiprime nearrings whose maximal right nearring of quotients are the centralizer nearrings determined by commutative fixed point free groups of automorphisms.

$ \sigma $-derivations on prime near-rings

The literatrue on near-rings contains a number of theorems asserting that certain conditions implying commutativity in rings imply multiplicative or additive commutativity in special classes of near-rings. H. E. Bell and G. Mason in [2] added to this body of results several commutativity theorems for near-rings admitting suitably-constrained derivations. In this paper we generalize some of their results to a subclass of prime near-rings admitting suitably-constrained $ \sigma $-derivations, where $ \sigma $ is an automorphibm of the prime near-ring.

On prime and semiprime near‐rings with derivations

Let N be a semiprime right near‐ring, A a subset of N such that 0 ∈ A and AN⫅A, and d a derivation of N The purpose of this paper is to prove that if d acts as a homomorphism on A or as an anti‐homomorphism on A, then d(A) = {0}.

Posner and herstein theorems for derivations of 3-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