scholarly journals Dependent Elements of Derivations on Semiprime Rings

2009 ◽  
Vol 2009 ◽  
pp. 1-6
Author(s):  
Faisal Ali ◽  
Muhammad Anwar Chaudhry
Keyword(s):  
Semiprime Ring ◽  
Essential Ideal ◽  
Semiprime Rings

We characterize dependent elements of a commuting derivationdon a semiprime ringRand investigate a decomposition ofRusing dependent elements ofd. We show that there exist idealsUandVofRsuch thatU⊕Vis an essential ideal ofR,U∩V={0},d=0onU,d(V)⊆V, anddacts freely onV.

scholarly journals On the symmetric algebra of quotients of a C*-algebra

1990 ◽  
Vol 32 (3) ◽  
pp. 377-379 ◽  
Author(s):  
Pere Ara
Keyword(s):  
Galois Theory ◽  
Semiprime Ring ◽  
Equivalence Classes ◽  
Crossed Products ◽  
Prime Rings ◽  
C Algebra ◽  
Essential Ideal ◽  
Module Homomorphism ◽  
Semiprime Rings ◽  
Ring Of Quotients

Let R be a semiprime ring (possibly without 1). The symmetric ring of quotients of R is defined as the set of equivalence classes of essentially defined double centralizers (ƒ, g) on R; see [1], [8]. So, by definition, ƒ is a left R-module homomorphism from an essential ideal I of R into R, g is a right R-module homomorphism from an essential ideal J of R into R, and they satisfy the balanced condition ƒ(x)y = xg(y) for x ∈ Iand y ∈ J. This ring was used by Kharchenko in his investigations on the Galois theory of semiprime rings [4] and it is also a useful tool for the study of crossed products of prime rings [7]. We denote the symmetric ring of quotients of a semiprime ring R by Q(R).

A Note on Differential Identities in Prime and Semiprime Rings

2020 ◽  
pp. 77-83
Author(s):  
Mohammad Shadab Khan ◽  
Mohd Arif Raza ◽  
Nadeemur Rehman
Keyword(s):  
Prime Ring ◽  
Semiprime Ring ◽  
Nonzero Ideal ◽  
Differential Identities ◽  
Standard Identity ◽  
Prime And Semiprime Rings ◽  
Semiprime Rings ◽  
Positive Integers

Let R be a prime ring, I a nonzero ideal of R, d a derivation of R and m, n fixed positive integers. (i) If (d ( r ○ s)(r ○ s) + ( r ○ s) d ( r ○ s)n - d ( r ○ s))m for all r, s ϵ I, then R is commutative. (ii) If (d ( r ○ s)( r ○ s) + ( r ○ s) d ( r ○ s)n - d (r ○ s))m ϵ Z(R) for all r, s ϵ I, then R satisfies s4, the standard identity in four variables. Moreover, we also examine the case when R is a semiprime ring.

Derivations of differentially semiprime rings

2019 ◽  
Vol 12 (05) ◽  
pp. 1950079
Author(s):  
Ahmad Al Khalaf ◽  
Iman Taha ◽  
Orest D. Artemovych ◽  
Abdullah Aljouiiee
Keyword(s):  
Commutative Ring ◽  
Semiprime Ring ◽  
Prime Rings ◽  
Commutative Case ◽  
Lie Ring ◽  
Lie Rings ◽  
Semiprime Rings ◽  
Torsion Free

Earlier D. A. Jordan, C. R. Jordan and D. S. Passman have investigated the properties of Lie rings Der [Formula: see text] of derivations in a commutative differentially prime rings [Formula: see text]. We study Lie rings Der [Formula: see text] in the non-commutative case and prove that if [Formula: see text] is a [Formula: see text]-torsion-free [Formula: see text]-semiprime ring, then [Formula: see text] is a semiprime Lie ring or [Formula: see text] is a commutative ring.

Additive n-commuting maps on semiprime rings

2019 ◽  
Vol 63 (1) ◽  
pp. 193-216
Author(s):  
Cheng-Kai Liu
Keyword(s):  
Unsolved Problem ◽  
Semiprime Ring ◽  
Affirmative Answer ◽  
Extended Centroid ◽  
Additive Map ◽  
Semiprime Rings ◽  
Ring Of Quotients ◽  
Functional Identities ◽  
Fixed Positive Integer ◽  
Commuting Maps

AbstractLet R be a semiprime ring with the extended centroid C and Q the maximal right ring of quotients of R. Set [y, x]1 = [y, x] = yx − xy for x, y ∈ Q and inductively [y, x]k = [[y, x]k−1, x] for k > 1. Suppose that f : R → Q is an additive map satisfying [f(x), x]n = 0 for all x ∈ R, where n is a fixed positive integer. Then it can be shown that there exist λ ∈ C and an additive map μ : R → C such that f(x) = λx + μ(x) for all x ∈ R. This gives the affirmative answer to the unsolved problem of such functional identities initiated by Brešar in 1996.

On commuting additive mappings on semiprime rings

2020 ◽  
pp. 2150079
Author(s):  
Siriporn Lapuangkham ◽  
Utsanee Leerawat
Keyword(s):  
Prime Ring ◽  
Semiprime Ring ◽  
Semiprime Rings ◽  
Additive Mappings

The main purpose of this paper is to describe the structure of a pair of additive mappings that are commuting on a semiprime ring. Furthermore, we prove that the existence of different commuting epimorphisms on a prime ring forces the ring to be commutative. Finally, we characterize additive mappings, which act as homomorphisms or antihomomorphisms on a semiprime ring.

scholarly journals A Commutativity theorem for semiprime rings

1980 ◽  
Vol 30 (1) ◽  
pp. 33-36
Author(s):  
Vishnu Gupta
Keyword(s):  
Positive Integer ◽  
Semiprime Ring ◽  
Commutativity Theorem ◽  
Semiprime Rings

AbstractIt is shown that if R is a semiprime ring with 1 satisfying the property that, for each x, y ∈ R, there exists a positive integer n depending on x and y such that (xy)k − xkyk is central for k = n,n+1, n+2, then R is commutative, thus generalizing a result of Kaya.

scholarly journals On Generalized ()-Derivations in Semiprime Rings

ISRN Algebra ◽  
2012 ◽  
Vol 2012 ◽  
pp. 1-7 ◽  
Author(s):  
Basudeb Dhara ◽  
Atanu Pattanayak
Keyword(s):  
Semiprime Ring ◽  
Generalized Derivation ◽  
Additive Mapping ◽  
Nonzero Ideal ◽  
Generalized Derivations ◽  
Semiprime Rings

Let be a semiprime ring, a nonzero ideal of , and , two epimorphisms of . An additive mapping is generalized -derivation on if there exists a -derivation such that holds for all . In this paper, it is shown that if , then contains a nonzero central ideal of , if one of the following holds: (i) ; (ii) ; (iii) ; (iv) ; (v) for all .

scholarly journals A note on a pair of derivations of semiprime rings

2004 ◽  
Vol 2004 (39) ◽  
pp. 2097-2102 ◽  
Author(s):  
Muhammad Anwar Chaudhry ◽  
A. B. Thaheem
Keyword(s):  
Semiprime Ring ◽  
Banach Algebras ◽  
Semiprime Rings

We study certain properties of derivations on semiprime rings. The main purpose is to prove the following result: letRbe a semiprime ring with centerZ(R), and letf,gbe derivations ofRsuch thatf(x)x+xg(x)∈Z(R)for allx∈R, thenfandgare central. As an application, we show that noncommutative semisimple Banach algebras do not admit nonzero linear derivations satisfying the above central property. We also show that every skew-centralizing derivationfof a semiprime ringRis skew-commuting.

Centralizing Mappings of Semiprime Rings

1987 ◽  
Vol 30 (1) ◽  
pp. 92-101 ◽  
Author(s):  
H. E. Bell ◽  
W. S. Martindale
Keyword(s):  
Nonempty Subset ◽  
Semiprime Ring ◽  
Prime Rings ◽  
Semiprime Rings

AbstractLet R be a ring with center Z, and S a nonempty subset of R. A mapping F from R to R is called centralizing on S if [x, F(x)] ∊ Z for all x ∊ S. We show that a semiprime ring R must have a nontrivial central ideal if it admits an appropriate endomorphism or derivation which is centralizing on some nontrivial one-sided ideal. Under similar hypotheses, we prove commutativity in prime rings.

GENERALIZED (α, β)*-DERIVATIONS AND RELATED MAPPINGS IN SEMIPRIME *-RINGS

2012 ◽  
Vol 05 (02) ◽  
pp. 1250015
Author(s):  
Mohammad Ashraf ◽  
Shakir Ali ◽  
Almas Khan
Keyword(s):  
Semiprime Ring ◽  
Semiprime Rings

Let R be a *-ring. Suppose α and β are the endomorphisms of R. In this paper we introduce the notions of generalized (α, β)*-derivation and symmetric generalized (α, β)*-biderivation. It is shown that if a semiprime *-ring admits a generalized (α, β)*-derivation F such that α is surjective then F maps the ring R into its center. Analogous results have been proved for symmetric generalized (α, β)*-biderivation.

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