Akhil Chandra Paul

• University of Rajshahi

We study and develop the concepts of homomorphism and anti-homomorphism to derive some important results in the theory of gamma rings. This article attempts to analyze some of the results of Ali et al. [1] in case of classical rings for extending those in the context of gamma rings. We establish a number of results related to automorphism, anti-aut...

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 centralizi...

In this paper, we study the orthogonality of two generalized derivations in semiprime G-rings. Some results are obtained in connection with ideals of semiprime G-rings and using left annihilator which is taken to be zero. GANIT J. Bangladesh Math. Soc.Vol. 39 (2019) 63-70

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...

p>In this paper we prove that, if U is a s-square closed Lie ideal of a 2-torsion free s-prime ring R and d: R(R is an additive mapping satisfying d(u²)=d(u)u+ud(u) for all uεU then d(uv)=d(u)v+ud(v) holds for all u,vεU GANIT J. Bangladesh Math. Soc. Vol. 36 (2016) 1-5</p

Let M be a 2-torsion free prime \({\varGamma }\)-ring with Z(M) as the center of M. In this paper, we prove the following: (i) If U is a Lie ideal of M and if \(d\not =0\) is a derivation of M such that \(d^2(U) = 0\), then \(U\subseteq Z(M)\); (ii) if \(U\not \subset Z(M)\) is a Lie ideal of M and \(d\not =0\) is a derivation of M, then \(Z(d(U))\...

The objective of this paper is to introduce the notion of a permuting skew tri-derivation on prime and semiprime \(\varGamma \)-rings. We prove that under certain conditions a prime \(\varGamma \)-ring is to be commutative by means of a nonzero permuting skew tri-derivation.

Let M be a 2-torsion free δ-prime T-ring satisfying the condition abc = abc for all a, b, c ϵ M and, ϵ T, I a δ-prime ideal of M and d a semiderivation associated with a function g which is surjective on I. In the paper we show some conditions on d, such that d = 0 or M is commutative.

We develop some significant results on k-derivations and generalized k-derivations of Nobusawa gamma rings to determine the commutativity of prime Nobusawa Γ-rings of characteristic not equal to 2 with generalized k-derivations. This article also shows that every Jordan k-homomorphism θ: M→N of a Nobusawa Γ1-ring M onto a 2-torsion free prime Nobusa...

Let M be a Г-ring and let D: M x M ->M be a symmetric bi-derivation with the trace d: M -> M denoted by d(x) = D(x, x) for all xεM. The objective of this paper is to prove some results concerning symmetric bi-derivation on prime and semiprime Г-rings. If M is a 2-torsion free prime Г-ring and D ≠ 0 be a symmetric bi-derivation with the trace d havi...

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

Let M be a 2-torsion free semiprime G-ring satisfying the condition aαbβc = aβbαc,∀a, b, c ͼM and α, β ͼГ. Let U be an admissible Lie ideal of M that is, uαu ͼ U,∀u ͼ U, α ͼG and U ͼZ(M), the centre of M. If d : M -> M is an additive mapping such that d is a Jordan derivation on U of M, then d is a derivation on U.GANIT J. Bangladesh Math. Soc.Vol....

Let U be a {\sigma}-square closed Lie ideal of a 2-torsion free {\sigma}-prime {\Gamma}-ring M. Let d{\neq}1 be an automorphism of M such that on U, on U, and there exists u_0 in Sa_{\sigma}(M) with M{\Gamma}u_0{\subseteq}U. Then, U{\subseteq}Z(M). By applying this result, we generalize the results of Oukhtite and Salhi respect to {\Gamma}-rings. F...

Let U be a non-zero square closed Lie ideal of a 2-torsion free prime ring R and f a generalized derivation of R with the associated derivation d of R. If f acts as a homomorphism and as an anti-homomorphism on U, then we prove that d = 0 or U € Z(R), the centre of R.GANIT J. Bangladesh Math. Soc.Vol. 35 (2015) 73-77

Let M be a 2-torsion free sigma-prime Gamma-ring and U be a non-zero sigma-square closed Lie ideal of M. If T : M -> M is an automorphism on U such that T not equal 1 and T sigma = sigma T on U, then we prove that U subset of Z (M). We also study the additive maps d : M -> M such that d(u alpha u) = 2u alpha d(u), where u is an element of U and alp...

Let M be a 2-torsion free -ring with involution I satisfying the condition for all and . The object of our paper is to show that every Jordan left-I-centralizer on a semiprime -ring with involution I, is a reverse left-I-centralizer. We solve some functional equations in prime and semiprime -rings with involution I by means of the above results. Mo...

Let M be a prime {\Gamma}-ring and let d be a derivation of M. If there exists a fixed integer n such that $(d(x){\alpha})^nd(x)

Let N be a prime G -near-ring and s be an automorphism on N . In this paper, we prove that if d is a s -derivation of N such that s d = d s with d 2 = 0, then d = 0. The composition of two derivations s and τ are considered and investigated the conditions that the derivation is a στ-derivation. © 2014 JSR Publications. ISSN: 2070-0237 (Print); 2070...

Let M be a prime ?-ring satisfying a certain assumption a?b?c = a?b?c for all a, b, c?M and ?, ???, and let I be an ideal of M. Assume that (D, d) is a generalized derivation of M and a?M. If D([x, a]?) = 0 or [D(x), a]? = 0 for all x?I, ? ? ?, then we prove that d(x) = p?[x, a]? for all x?I, ?, ? ? ? or a?Z(M) (the centre of M), where p belongs C(...

Let M be a G-ring. If M satisfies the condition (*) xaybz = xbyaz for all x, y, zÎM, a, bÎG, then we investigate commutativity of prime G-rings satisfying certain identities involving left centralizer. Keywords: Prime G-ring; Derivation; Generalized derivation; Left centralizer. © 2014 JSR Publications. ISSN: 2070-0237 (Print); 2070-0245 (Online)....

In this paper we prove that any completely prime G -ring M satisfying the condition aabb c = ab ba c (a, b, cÎM anda ,b ÎG) with nonzero derivation, is a commutative integral G -domain if its characteristic is not two. We also show that if the characteristic of M is 2 the G -ring M is either commutative or is an order in a simple 4-dimensional alge...

The object of this paper is to introduce a permuting tri-derivation in a G-near-ring. We obtain the conditions for a prime G-near-ring to be a commutative G-ring. Keywords : Gamma- near-ring; Prime Gamma-near-ring; Commutative Gamma-ring; Permuting tri-derivation. © 2013 JSR Publications. ISSN: 2070-0237 (Print); 2070-0245 (Online). All rights rese...

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.

Let N be a prime Lnear-ring with the center Z(N). The objective of this paper is to study derivations on N. We prove two results: (a) Let N be 2-torsion free and let D1 and D2 be derivations on N such that D1D2 is also a derivation. Then D1 = 0 or D2 = 0 if and only if [D1(x),D2(y)]α = 0 for all x, y γ N, α (b) Let n be an integer greater than 1, N...

Let N be a non zero-symmetric left ?-near-ring. If N is a prime ?-near-ring with nonzero derivations D1 and D2 such that D1(x)?D2(y) = D2(x)?D1(y) for every x, y?N and ???, then we prove that N is an abelian ?-near-ring. Again if N is a 2-torsion free prime ?-near-ring and D1 and D2 are derivations satisfying D1(x)?D2(y) = D2(x)?D1(y) for every x,...

This article defines k -endomorphism and anti- k -endomorphism on Γ N -rings, and uses the concept of k -derivation of Γ N -rings. Considering M as a semiprime Γ N -ring and d as a k -derivation of M , it aims to prove that (i) if d acts as a k -endomorphism on M such that M Γ M = M and xk (α) x =0 for all x ∈ M and α∈ Γ, then d =0; and (ii) if d i...

We study some properties of permuting tri-derivations on semiprime G-rings with a certain assumption. Let M be a 3-torsion free semiprime G-ring satisfying a certain assumption and let I be a non-zero ideal of M. Suppose that there exists a permuting tri-derivation D: M×M´M ? M such that d is an automorphism commuting on I and also d is a trace of...

Let M be a semiprime G -ring satisfying an assumption xa yb z = xb ya z for all x, y, z ? M , a, b?G . In this paper, we prove that a mapping T : M ? M is a centralizer if and only if it is a centralizing left centralizer. We also show that if T and S are left centralizers of M such that T ( x ) a x + x a S ( x )? Z ( M ) (the center of M ) for all...

Let M be a prime ?-ring satisfying a certain assumption (*). An additive mapping f : M ? M is a semi-derivation if f(x?y) = f(x)?g(y) + x?f(y) = f(x)?y + g(x)?f(y) and f(g(x)) = g(f(x)) for all x, y?M and ? ? ?, where g : M?M is an associated function. In this paper, we generalize some properties of prime rings with semi-derivations to the prime &G...

Let M be a prime G-ring and let I be a nonzero ideal of M. Suppose that D: M ® M is a nonzero generalized derivation with associated derivation d : M ® M. Then we prove the following: (i) If D acts as a homomorphism on I, then either d = 0 on M or M is commutative.(ii) If M satisfies the assumption (*) (see below), and if D acts as an anti-homomorp...

The notion of a gamma ring is a generalization of the concept of a classical ring. This attempt characterizes certain gamma rings with various types of k-derivations and k-homomorphisms. We determine the commutativity of prime gamma rings of characteristic not equal to 2 and 3 with k-derivations, left (and right) k-derivations and generalized k-der...

Let M be a 2-torsion free G-ring satisfying an assumption and let s , t be centralizing epimorphisms on M . Let f and g be ( s , t )-derivations on M such that f ( x ) αx + xαg ( x ) = 0 for all x I M , α IG. Then we prove that f ( u ) β [ x , y ] α = g ( u ) β [ x, y ] α = 0 for all x, y, u I M , α,β I G and f, g map M into its center. Keywords ....

Let M be 2 and 3 torsion-free left sΓ-unital Γ-rings. Let D: M ×M ×M ® M be a permuting tri-additive mapping with the trace d(x) = D(x,x,x). Let σ: M ® M be an endomorphism and τ: M ® M an epimorphism. The objective of this paper is to prove the following: a) If d is (σ,τ)-skew commuting on M, then D = 0; b) If d is (τ,τ)-skew-centralizing on M, th...

From the very definition, it follows that every Jordan k -derivation of a gamma ring M is, in general, not a k -derivation of M . In this article, we establish its generalization by considering M as a 2-torsion free semiprime ? N -ring (Nobusawa gamma ring). We also show that every Jordan k -derivation of a 2-torsion free completely semiprime ? N -...

We first define k-isomorphism, anti-k-isomorphism and Jordan k-isomorphism of Nobusawa gamma rings and then develop some useful consequences to prove that every Jordan k-isomorphism of a Nobusawa gamma ring onto a 2-torsion free prime Nobusawa gamma ring is either a k-isomorphism or an anti-k-isomorphism. Next we are to show that the similar result...

This article is based on some derivations of certain gamma rings. By giving the definitions of k-derivation and Jordan k-derivation of a gamma ring as well as that of certain gamma rings, some results related to these concepts are developed. Clearly, every k-derivation of a Γ-ring M is also a Jordan k-derivation of M. But, the converse is not true...

With the notions of a left k-derivation and a Jordan left k-derivation of a Γ-ring we construct some important results relating to them in a concrete manner. In this article, we show that under a suitable condition every nonzero Jordan left k-derivation d of a 2-torsion free completely prime Γ-ring M induces the commutativity of M, and accordingly,...

Every Jordan generalized k-derivation of a Γ-ring is not a generalized k-derivation of the same. In this paper, we show that under some conditions every Jordan generalized k-derivation of a 2-torsion free completely semiprime ΓN-ring is a generalized k-derivation by developing a number of results relating to these derivations of certain Γ-rings.

In this article, we develop some important results relating to the concepts of generalized k -derivation and Jordan generalized k-derivation of certain gamma rings. Though every generalized k-derivation of a gamma ring M is obviously a Jordan generalized k-derivation of M, but the converse statement is in general not true. Here we prove that every...

We use the concept of k-derivation of a gamma ring to develop a number of important results on k-derivations of prime Nobusawa gamma rings. Especially, we characterize some significant consequences of the commutativity of prime Nobusawa gamma rings of characteristic not equal to 2 and 3 with k-derivations and also with the composition of two k-deri...