Semiprime Gamma Research Articles

Several types of derivations on prime and semiprime gamma rings

In this paper the notion of asymmetric skew 4-derivation of prime and semiprime Γ-ring is presented and studied. Therefore we proved that; presume Ň be a 3,2-torsion free non commutative prime Γ-rings accomplishing a certain assumption and L be admissible Lie ideal of Ň; Let : Ň4→Ň is a symmetric 4-derivation. If f is a trace of such that [f(s),s]σ = 0; for each one s ∈ L, σ ∈ Γ therefore Ῥ = 0,also we prove that; if Ň be a 3,2−torsion free non commutative semiprime Γ-rings accomplishing a certain assumption and L be admissible Lie ideal of Ň. Presume a is an automorphism of Ň and : Ň4→Ň is a symmetric skew 4-derivation associated with a. Presume that the trace function f is commuting on L such that [f(s), a(s)]σ ∈ Z for each s ∈ L, σ ∈ Γ, where f(L) ⸦ L, therefore [f(s), a(s)]σ = 0 for eachs ∈ L, σ ∈ Γ,the quantity demand and investment in research and development, while the other model focuses on a more realistic relationship between the quantity demand and the price.

Some Characterizations on Actions of Generalized Derivations AsHomomorphisms And Anti-Homomorphisms In SemiprimeGamma Rings

Let I be a non-zero left ideal of a r-ring M satisfying the condition alpha alpha b beta c= a beta b alpha c for all a,b,c, e M and alpha, beta e r. We show that M contains a non-trivial central ideal if M is semiprime which admits an appropriate non-zero derivations on I, and also that M is commutative if M is prime admitting a non-zero centralizing derivation on I. We next give some characterizations when non-zero generalized derivations act as homomorphisms and as anti-homomorphisms on some non-zero left or two-sided ideals of semiprime gamma rings, somewhere of prime gamma rings also, satisfying the above condition.GANIT J. Bangladesh Math. Soc. 42.1 (2022) 025- 034

On Prime and Semiprime Gamma Rings with Symmetric Gamma n-Centralizers

In this paper, we assume that M is Γ-ring then we will present the definition of symmetric Γ-n-centralizers and the definition of Jordan Γ-n-centralizers of M. After that, we try to find the commutativity of a prime Γ-ring that has a left Γ-n-centralizers of M when it satisfies certain identities. Finally, we find the relationship between Jordan Γ-n-centralizers and symmetric Γ-n-centralizers when M is semiprime Γ-ring.

Symmetric Reverse Gamma *-4-Centralizers on SemiprimeGamma Rings with Involution

Let M be a -ring with involution . In this paper , we will introduce the concept of symmetric left(right) reverse *-4-centralizer of M . Then, we proved that the 4-additive mapping T:MxMxMxMM is a reverse *-4-centralizer of M under certain conditions .

Orthogonal generalized higher derivations in semi prime gamma near-rings

In this paper, we present some result concerning to generalized higher derivations on a semi prime gamma-near – rings. Also, orthogonality is defined on to generalization of higher derivations. In addition to the proof of theories on this topic.

Inner Derivations on Semiprime Gamma Rings

We study inner derivations and generalized inner derivations in semiprime Γ-rings to develop some important results. If f and g are inner derivations of a semiprime Γ-ring M satisfying the equation for all , then we show that . This equation produces a number of results on generalized inner derivations as well.
 GANIT J. Bangladesh Math. Soc.Vol. 39 (2019) 101-110

On Semiprime Gamma Near-Rings with Perpendicular Generalized 3-Derivations

In this paper , we introduce the notion of perpendicular generalized 3-derivations in semiprime gamma near-rings and present several necessary and sufficient conditions for generalized 3-derivations on semiprime gamma near-rings to be perpendicular .

Commutators in semiprime gamma rings

We characterize connections between associative, Lie and Jordan structures of semiprime gamma rings.

On derivations of prime and semi-prime Gamma rings

The concept of $\Gamma$-ring is a generalization of ring. Twoimportant classes of $\Gamma$-rings are prime and semi-prime$\Gamma$-rings. In this paper, we consider the concept of derivations on prime and semi-prime $\Gamma$-rings and we study some of their properties.

Jordan Derivations on Completely Semiprime Gamma–Rings

In this paper we prove that under a suitable condition every Jordan derivation on a 2-torsion free completely semiprime ?-ring is a derivation.GANIT J. Bangladesh Math. Soc.Vol. 34 (2014) 21-26

On Jordan left-I-centralizers of prime and semiprime gamma rings with involution

Let M be a 2-torsion free Γ-ring with involution I satisfying the condition xαyβz=xβyαz for all x,y,z∈M and α,β∈Γ. The object of our paper is to show that every Jordan left-I-centralizer on a semiprime Γ-ring with involution I, is a reverse left-I-centralizer. We solve some functional equations in prime and semiprime Γ-rings with involution I by means of the above results. Moreover, we discuss some more related results.

Left centralizers of semiprime gamma rings with involution

The purpose of this article is to define notions of I-involution on Γ-rings, existence and prove some interesting results: (i) Let M be a 2-torsion free semiprime Γ-ring with I-involution and satisfying a certain assumption. If T : M → M is a Jordan left centralizer on M , then T is a left centralizer. (ii) Let M be a 2-torsion free semiprime Γ-ring with I-involution and T : M → M an additive mapping such that T (xαI(x )+ I(x)αx )= T (x)αI(x )+ I(x)αT (x)

SEMIPRIME GAMMA RINGS WITH ORTHOGONAL REVERSE DERIVATIONS

In this paper, the definition of orthogonal reverse derivations is given. Some characterizations of semiprime gamma rings are obtained by means of orthogonal reverse derivations. We also investigate conditions for two reverse derivations to be orthogonal.

Generalized Derivations in Semiprime Gamma Rings

LetMbe a 2-torsion-free semiprimeΓ-ring satisfying the conditionaαbβc=aβbαcfor alla,b,c∈M, α,β∈Γ, and letD:M→Mbe an additive mapping such thatD(xαx)=D(x)αx+xαd(x)for allx∈M, α∈Γand for some derivationdofM. We prove thatDis a generalized derivation.

Dependent Elements in Prime and Semiprime Gamma Rings

Dependent Elements in Prime and Semiprime Gamma Rings

REGULARITY OF THE GENERALIZED CENTROID OF SEMI-PRIME GAMMA RINGS

The aim of this note is to study properties of the generalized centroid of the semi-prime gamma rings. Main results are the following theorems: (1) Let M be a semi-prime <TEX>$\Gamma$</TEX>-ring and Q a quotient <TEX>$\Gamma$</TEX>-ring of M. If W is a non-zero submodule of the right (left) M-module Q, then <TEX>$W\Gamma$W</TEX> <TEX>$\neq</TEX> 0. Furthermore Q is a semi-prime <TEX>$\Gamma$</TEX>-ring. (2) Let M be a semi-prime <TEX>$\Gamma$</TEX>-ring and <TEX>$C_{{Gamma}$</TEX> the generalized centroid of M. Then <TEX>$C_{\Gamma}$</TEX> is a regular <TEX>$\Gamma$</TEX>-ring. (3) Let M be a semi-prime <TEX>$\Gamma$</TEX>-ring and <TEX>$C_{\gamma}$</TEX> the extended centroid of M. If <TEX>$C_{\gamma}$</TEX> is a <TEX>$\Gamma$</TEX>-field, then the <TEX>$\Gamma$</TEX>-ring M is a prime <TEX>$\Gamma$</TEX>-ring.