On Jordan Structure in Semiprime Rings

1976 ◽  
Vol 28 (5) ◽  
pp. 1067-1072 ◽  
Author(s):  
Ram Awtar
Keyword(s):  
Semiprime Ring ◽  
Nonzero Ideal ◽  
Lie Ideal ◽  
Additive Subgroup ◽  
Jordan Structure ◽  
Semiprime Rings

A remarkable theorem of Herstein [1, Theorem 2] of which we have made several uses states: If R is a semiprime ring of characteristic different from 2 and if U is both a Lie ideal and a subring of R then either U ⊂ Z (the centre of R) or U contains a nonzero ideal of R. In a recent paper [3] Herstein extends the above mentioned result significantly and has proved that if R is a semiprime ring of characteristic different from 2 and V is an additive subgroup of R such that [V, U] ⊂ V, where U is a Lie ideal of R, then either [V, U] = 0 or V ⊃ [M, R] ≠ 0 where M is an ideal of R. In this paper our object is to prove the following.

A study of derivable mappings of semiprime rings

2018 ◽  
Vol 7 (1-2) ◽  
pp. 19-26
Author(s):  
Gurninder S. Sandhu ◽  
Deepak Kumara
Keyword(s):  
Associative Ring ◽  
Semiprime Ring ◽  
Nonzero Ideal ◽  
Lie Ideal ◽  
Commutativity Theorems ◽  
Semiprime Rings

Throughout this note, \(R\) denotes an associative ring and \(C(R)\) be the center of \(R\). In this paper, it isproved that a non-central Lie ideal \(L\) of a semiprime ring \(R\) contains a nonzero ideal of \(R\) and this result isused to obtain several commutativity theorems of \(R\) involving multiplicative derivations. Moreover, someresults on one-sided ideals of \(R\) are given.

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.

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 On n-generalized commutators and Lie ideals of rings

2021 ◽  
pp. 2250221
Author(s):  
Peter V. Danchev ◽  
Tsiu-Kwen Lee
Keyword(s):  
Positive Integer ◽  
Prime Ring ◽  
Classical Result ◽  
Associative Ring ◽  
Nonzero Ideal ◽  
Lie Ideal ◽  
Additive Subgroup ◽  
Simple Ring ◽  
Noncommutative Polynomials ◽  
Generalized Commutator

Let [Formula: see text] be an associative ring. Given a positive integer [Formula: see text], for [Formula: see text] we define [Formula: see text], the [Formula: see text]-generalized commutator of [Formula: see text]. By an [Formula: see text]-generalized Lie ideal of [Formula: see text] (at the [Formula: see text]th position with [Formula: see text]) we mean an additive subgroup [Formula: see text] of [Formula: see text] satisfying [Formula: see text] for all [Formula: see text] and all [Formula: see text], where [Formula: see text]. In the paper, we study [Formula: see text]-generalized commutators of rings and prove that if [Formula: see text] is a noncommutative prime ring and [Formula: see text], then every nonzero [Formula: see text]-generalized Lie ideal of [Formula: see text] contains a nonzero ideal. Therefore, if [Formula: see text] is a noncommutative simple ring, then [Formula: see text]. This extends a classical result due to Herstein [Generalized commutators in rings, Portugal. Math. 13 (1954) 137–139]. Some generalizations and related questions on [Formula: see text]-generalized commutators and their relationship with noncommutative polynomials are also discussed.

scholarly journals (U. M)-derivations in completely semiprime ?-rings

2016 ◽  
Vol 51 (1) ◽  
pp. 69-74
Author(s):  
MM Rahman ◽  
AC Paul
Keyword(s):  
Semiprime Ring ◽  
Subject Classification ◽  
Lie Ideal ◽  
Semiprime Rings ◽  
And Mathematics

The objective of this paper is to extend and generalize some results of (Rahman and Paul, 2014) in completely semiprime ?- rings. We prove that, if is an admissible Lie ideal of a completely semiprime ?-ring and d is a (U, M) -derivation of then d(u?v)=d(u)?v+u?d(v) for all u, v ? U and ? ? ?Mathematics Subject Classification: 13N15, 16W10, 17C50Bangladesh J. Sci. Ind. Res. 51(1). 69-74, 2016

On Prime and Semiprime Rings with Derivations

2006 ◽  
Vol 13 (03) ◽  
pp. 371-380 ◽  
Author(s):  
Nurcan Argaç
Keyword(s):  
Prime Ring ◽  
Nonempty Subset ◽  
Semiprime Ring ◽  
Nonzero Ideal ◽  
Prime Rings ◽  
Prime And Semiprime Rings ◽  
Semiprime Rings ◽  
Torsion Free

Let R be a ring and S a nonempty subset of R. A mapping f: R → R is called commuting on S if [f(x),x] = 0 for all x ∈ S. In this paper, firstly, we generalize the well-known result of Posner related to commuting derivations on prime rings. Secondly, we show that if R is a semiprime ring and I is a nonzero ideal of R, then a derivation d of R is commuting on I if one of the following conditions holds: (i) For all x, y ∈ I, either d([x,y]) = [x,y] or d([x,y]) = -[x,y]. (ii) For all x, y ∈ I, either d(x ◦ y) = x ◦ y or d(x ◦ y) = -(x ◦ y). (iii) R is 2-torsion free, and for all x, y ∈ I, either [d(x),d(y)] = d([x,y]) or [d(x),d(y)] = d([y,x]). Furthermore, if d(I) ≠ {0}, then R has a nonzero central ideal. Finally, we introduce the notation of generalized biderivation and prove that every generalized biderivation on a noncommutative prime ring is a biderivation.

scholarly journals Remarks on derivations on semiprime rings

1992 ◽  
Vol 15 (1) ◽  
pp. 205-206 ◽  
Author(s):  
Mohamad Nagy Daif ◽  
Howard E. Bell
Keyword(s):  
Semiprime Ring ◽  
Nonzero Ideal ◽  
Semiprime Rings

We prove that a semiprime ringRmust be commutative if it admits a derivationdsuch that (i)xy+d(xy)=yx+d(yx)for allx,yinR, or (ii)xy−d(xy)=yx−d(yx)for allx,yinR. In the event thatRis prime, (i) or (ii) need only be assumed for allx,yin some nonzero ideal ofR.

scholarly journals Derivations on Lie ideals of completely semiprime ? -rings

2016 ◽  
Vol 27 (1) ◽  
pp. 51-61
Author(s):  
MM Rahman ◽  
AC Paul
Keyword(s):  
Semiprime Ring ◽  
Jordan Derivation ◽  
Lie Ideal ◽  
Semiprime Rings ◽  
Torsion Free

The aim of the paper is to prove the following theorem concerning a class of ? -rings. LetM be a 2-torsion free completely semiprime ? -ring satisfying the condition a?b?c = a?b?c, for all a,b,c?M and ? ,? ?? ; U be an admissible Lie ideal ofM . If d :M ?M is a Jordan derivation on U of M , then d is a derivation on U of M .Bangladesh J. Sci. Res. 27(1): 51-61, June-2014

scholarly journals A note on semiprime rings with derivation

1997 ◽  
Vol 20 (2) ◽  
pp. 413-415 ◽  
Author(s):  
Motoshi Hongan
Keyword(s):  
Semiprime Ring ◽  
Nonzero Ideal ◽  
Semiprime Rings ◽  
Torsion Free

LetRbe a2-torsion free semiprime ring,Ia nonzero ideal ofR,Zthe center ofRandD:R→Ra derivation. Ifd[x,y]+[x,y]∈Zord[x,y]−[x,y]∈Zfor allx,y∈I, thenRis commutative.

scholarly journals Remarks on Generalized Derivations in Prime and Semiprime Rings

2010 ◽  
Vol 2010 ◽  
pp. 1-6 ◽  
Author(s):  
Basudeb Dhara
Keyword(s):  
Prime Ring ◽  
Semiprime Ring ◽  
Generalized Derivation ◽  
Additive Mapping ◽  
Nonzero Ideal ◽  
Generalized Derivations ◽  
Prime And Semiprime Rings ◽  
Semiprime Rings

LetRbe a ring with centerZandIa nonzero ideal ofR. An additive mappingF:R→Ris called a generalized derivation ofRif there exists a derivationd:R→Rsuch thatF(xy)=F(x)y+xd(y)for allx,y∈R. In the present paper, we prove that ifF([x,y])=±[x,y]for allx,y∈IorF(x∘y)=±(x∘y)for allx,y∈I, then the semiprime ringRmust contains a nonzero central ideal, providedd(I)≠0. In caseRis prime ring,Rmust be commutative, providedd≠0. The cases (i)F([x,y])±[x,y]∈Zand (ii)F(x∘y)±(x∘y)∈Zfor allx,y∈Iare also studied.

Sign in / Sign up

Export Citation Format

Share Document