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.

Generalized Derivations Having the Same Power Values with Left Multiplications

2013 ◽  
Vol 20 (03) ◽  
pp. 369-382 ◽  
Author(s):  
Xiaowei Xu ◽  
Jing Ma ◽  
Fengwen Niu
Keyword(s):  
Positive Integer ◽  
Left Ideal ◽  
Prime Ring ◽  
Generalized Derivation ◽  
Nonzero Ideal ◽  
Lie Ideal ◽  
Extended Centroid ◽  
Generalized Derivations ◽  
Ring Of Quotients ◽  
Fixed Positive Integer

Let R be a prime ring with extended centroid C, maximal right ring of quotients U, a nonzero ideal I and a generalized derivation δ. Suppose δ(x)n =(ax)n for all x ∈ I, where a ∈ U and n is a fixed positive integer. Then δ(x)=λax for some λ ∈ C. We also prove two generalized versions by replacing I with a nonzero left ideal [Formula: see text] and a noncommutative Lie ideal L, respectively.

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 A THEOREM ON DERIVATIONS ON PRIME RINGS

2011 ◽  
Vol 91 (2) ◽  
pp. 219-229
Author(s):  
KUN-SHAN LIU
Keyword(s):  
Positive Integer ◽  
Prime Ring ◽  
Nonzero Ideal ◽  
Prime Rings ◽  
Additive Map ◽  
Fixed Positive Integer

AbstractLet R be a prime ring, let I be a nonzero ideal of R and let n be a fixed positive integer. We prove that if the characteristic of R is either 0 or a prime p that is greater than 2n, then an additive map d that satisfies d(xn+1)=∑ nj=0xn−jd(x)xj for all x∈I must be a derivation.

Generalized Skew Derivations and Nilpotent Values on Lie Ideals

2019 ◽  
Vol 26 (04) ◽  
pp. 589-614
Author(s):  
Vincenzo De Filippis ◽  
Onofrio Mario Di Vincenzo
Keyword(s):  
Positive Integer ◽  
Prime Ring ◽  
Quotient Ring ◽  
Lie Ideal ◽  
Extended Centroid ◽  
Martindale Quotient Ring ◽  
Fixed Positive Integer

Let R be a prime ring of characteristic different from 2 and 3, Qr be its right Martindale quotient ring and C be its extended centroid. Suppose that F and G are generalized skew derivations of R, L a non-central Lie ideal of R and n ≥ 1 a fixed positive integer. Under appropriate conditions we prove that if (F(x)x – xG(x))n = 0 for all x ∈ L, then one of the following holds: (a) there exists c ∈ Qr such that F(x) = xc and G(x) = cx; (b) R satisfies s4 and there exist a, b, c ∈ Qr such that F(x) = ax + xc, G(x) = cx + xb and (a − b)2 = 0.

A note on automorphisms of lie ideals in prime rings

2018 ◽  
Vol 68 (5) ◽  
pp. 1223-1229 ◽  
Author(s):  
Bijan Davvaz ◽  
Mohd Arif Raza
Keyword(s):  
Positive Integer ◽  
Prime Ring ◽  
Lie Ideal ◽  
Prime Rings ◽  
Standard Identity ◽  
Identity Automorphism ◽  
Fixed Positive Integer

Abstract In the present paper, we prove that a prime ring R with center Z satisfies s4, the standard identity in four variables if R admits a non-identity automorphism σ such that (uσ,u]vσ+vσ[uσ,u])n∈Z for all u,v in some non-central Lie ideal L of R whenever either char(R)>n or char(R)=0, where n is a fixed positive integer.

scholarly journals ON SUPERNILPOTENT NONSPECIAL RADICALS

2008 ◽  
Vol 78 (1) ◽  
pp. 107-110
Author(s):  
HALINA FRANCE-JACKSON
Keyword(s):  
Prime Ring ◽  
Jacobson Radical ◽  
Minimal Ideal ◽  
Negative Answer ◽  
Nonzero Ideal ◽  
Subdirectly Irreducible ◽  
Lower Radical ◽  
Simple Ring ◽  
Factor Ring ◽  
Semisimple Class

AbstractLet ρ be a supernilpotent radical. Let ρ* be the class of all rings A such that either A is a simple ring in ρ or the factor ring A/I is in ρ for every nonzero ideal I of A and every minimal ideal M of A is in ρ. Let $\mathcal {L}\left ( \rho ^{\ast }\right ) $ be the lower radical determined by ρ* and let ρφ denote the upper radical determined by the class of all subdirectly irreducible rings with ρ-semisimple hearts. Le Roux and Heyman proved that $\mathcal {L}\left ( \rho ^{\ast }\right ) $ is a supernilpotent radical with $\rho \subseteq \mathcal {L}\left ( \rho ^{\ast }\right ) \subseteq \rho _{\varphi }$ and they asked whether $\mathcal {L} \left ( \rho ^{\ast }\right ) =\rho _{\varphi }$ if ρ is replaced by β, ℒ , 𝒩 or 𝒥 , where β, ℒ , 𝒩 and 𝒥 denote the Baer, the Levitzki, the Koethe and the Jacobson radical, respectively. In the present paper we will give a negative answer to this question by showing that if ρ is a supernilpotent radical whose semisimple class contains a nonzero nonsimple * -ring without minimal ideals, then $\mathcal {L}\left ( \rho ^{\ast }\right ) $ is a nonspecial radical and consequently $\mathcal {L}\left ( \rho ^{\ast }\right ) \neq \rho _{\varphi }$. We recall that a prime ring A is a * -ring if A/I is in β for every $0\neq I\vartriangleleft A$.

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.

On prime right alternative rings with commutators in the left nucleus

1994 ◽  
Vol 50 (2) ◽  
pp. 287-298
Author(s):  
Erwin Kleinfeld ◽  
Harry F. Smith
Keyword(s):  
Associative Ring ◽  
Nonzero Ideal ◽  
Alternative Ring ◽  
Simple Ring ◽  
Characteristic 2 ◽  
Right Alternative Ring

A ring is called s–prime if the 2-sided annihilator of a nonzero ideal must be zero. In particular, any simple ring or prime (—1, 1) ring is s–prime. Also, a nonzero s–prime right alternative ring, with characteristic ≠ 2, cannot be right nilpotent. Let R be a right alternative ring with commutators in the left nucleus. Then R is associative in the following cases: (1) R is prime, with characteristic ≠ 2, and has an idempotent e ≠ 1 such that (e, e, R) = 0. (2) R is an algebra over a commutative-associative ring with 1/6, and R is either s–prime, or R is prime and locally (—1,1).

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.

Extensions of Rings Having McCoy Condition

2009 ◽  
Vol 52 (2) ◽  
pp. 267-272 ◽  
Author(s):  
Muhammet Tamer Koşan
Keyword(s):  
Positive Integer ◽  
Associative Ring ◽  
Nonzero Element ◽  
Ring Extensions

AbstractLet R be an associative ring with unity. Then R is said to be a right McCoy ring when the equation f (x)g(x) = 0 (over R[x]), where 0 ≠ f (x), g(x) ∈ R[x], implies that there exists a nonzero element c ∈ R such that f (x)c = 0. In this paper, we characterize some basic ring extensions of right McCoy rings and we prove that if R is a right McCoy ring, then R[x]/(xn) is a right McCoy ring for any positive integer n ≥ 2.

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