scholarly journals SOME RESULTS ON LIE IDEALS WITH SYMMETRIC REVERSE BI-DERIVATIONS IN SEMIPRIME RINGS I

2021 ◽  
pp. 309
Author(s):  
Emine Koç Sögütcü ◽  
Öznur Gölbaşı
Keyword(s):  
Commutative Ring ◽  
Semiprime Ring ◽  
Lie Ideal ◽  
R And D ◽  
Semiprime Rings ◽  
Square Closed Lie Ideal

Let R be a semiprime ring, U a square-closed Lie ideal of R and D : R R ! R a symmetric reverse bi-derivation and d be the trace of D: In the present paper, we shall prove that R commutative ring if any one of the following holds: i) d(U) = (0); ii)d(U) Z; iii)[d (x) ; y] 2 Z; iv)d(x)oy 2 Z; v)d ([x; y])[d(x); y] 2 Z; vi)d (x y)(d(x)y) 2 Z; vii)d ([x; y])d(x)y 2 Z viii)d (x y) [d(x); y] 2 Z; ix)d(x) y [d(y); x] 2 Z; x)d([x; y]) (d(x) y) [d(y); x] 2 Z xi)[d(x); y] [g(y); x] 2 Z; for all x; y 2 U; where G : R R ! R is symmetric reverse bi-derivations such that g is the trace of

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.

scholarly journals Generalized Higher Derivations on Lie Ideals of Triangular Algebras

2017 ◽  
Vol 25 (1) ◽  
pp. 35-53
Author(s):  
Mohammad Ashraf ◽  
Nazia Parveen ◽  
Bilal Ahmad Wani
Keyword(s):  
Commutative Ring ◽  
Lie Ideal ◽  
Additive Subgroup ◽  
Triangular Algebra ◽  
Higher Derivation ◽  
Square Closed Lie Ideal ◽  
Triangular Algebras

Abstract Let be the triangular algebra consisting of unital algebras A and B over a commutative ring R with identity 1 and M be a unital (A; B)-bimodule. An additive subgroup L of A is said to be a Lie ideal of A if [L;A] ⊆ L. A non-central square closed Lie ideal L of A is known as an admissible Lie ideal. The main result of the present paper states that under certain restrictions on A, every generalized Jordan triple higher derivation of L into A is a generalized higher derivation of L into A.

scholarly journals Jordan Derivations on Lie Ideals of ?-Prime Rings

2017 ◽  
Vol 36 ◽  
pp. 1-5
Author(s):  
Akhil Chandra Paul ◽  
Md Mizanor Rahman
Keyword(s):  
Prime Ring ◽  
Additive Mapping ◽  
Lie Ideal ◽  
Prime Rings ◽  
R And D ◽  
Square Closed Lie Ideal ◽  
Torsion Free

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 u?U then d(uv)=d(u)v+ud(v) holds for all  u,v?UGANIT J. Bangladesh Math. Soc.Vol. 36 (2016) 1-5

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

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.

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

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 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 Generalized Derivations on Power Values of Lie Ideals in Prime and Semiprime Rings

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Vincenzo De Filippis ◽  
Nadeem UR Rehman ◽  
Abu Zaid Ansari
Keyword(s):  
Lie Ideal ◽  
Generalized Derivations ◽  
Fixed Integer ◽  
Free Ring ◽  
Prime And Semiprime Rings ◽  
Semiprime Rings ◽  
Positive Integers ◽  
Torsion Free

LetRbe a 2-torsion free ring and letLbe a noncentral Lie ideal ofR, and letF:R→RandG:R→Rbe two generalized derivations ofR. We will analyse the structure ofRin the following cases: (a)Ris prime andF(um)=G(un)for allu∈Land fixed positive integersm≠n; (b)Ris prime andF((upvq)m)=G((vrus)n)for allu,v∈Land fixed integersm,n,p,q,r,s≥1; (c)Ris semiprime andF((uv)n)=G((vu)n)for allu,v∈[R,R]and fixed integern≥1; and (d)Ris semiprime andF((uv)n)=G((vu)n)for allu,v∈Rand fixed integern≥1.

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.

Sign in / Sign up

Export Citation Format

Share Document