Derivations Cocentralizing Polynomials

Lee, Tsiu-Kwen; Shiue, Wen-Kwei

doi:10.11650/twjm/1500407017

Cited by 41 publications

(12 citation statements)

References 8 publications

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“…, x n ) 2 is central-valued on R or char(R) = 2 and R satisfies the standard identity s 4 of degree 4. Later in [2] Lee and Shiue obtained the same result by involving polynomials instead of multilinear polynomials. An approach that can be used in studying P (d, g, S) is to examine the size of P (d, g, S) and a reasonable criterion for studying the size of P (d, g, S) is to examine its left annihilator L P = {x ∈ R, xt = 0∀t ∈ P (d, g, S)} and its centralizer C P = {x ∈ R, [x, t] = 0∀t ∈ P (d, g, S)}: if P (d, g, S) is rather large we would expect that L P = 0 and C P = Z(R).…”

Section: Introductionmentioning

confidence: 72%

Cocentralizing and vanishing derivations on multilinear polynomials in prime rings

Filippis

2009

Sib Math J

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Let R be a prime ring of characteristic different from 2 and extended centroid C and let f (x 1 , . . . , x n ) be a multilinear polynomial over C not central-valued on R, while δ is a nonzero derivation of R. Suppose that d and g are derivations of R such that δ(d (f (r 1 , . . . , r n ))f (r 1 , . . . , r n ) − f (r 1 , . . . , r n )g(f (r 1 , . . . , r n ))) = 0 for all r 1 , . . . , r n ∈ R. Then d and g are both inner derivations on R and one of the following holds:(1) d = g = 0;(2) d = −g and f (x 1 , . . . , x n ) 2 is central-valued on R.

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Section: Introductionmentioning

confidence: 72%

Cocentralizing and vanishing derivations on multilinear polynomials in prime rings

Filippis

2009

Sib Math J

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“…Continuing this line of investigation, in [20] Lee and Shiue extended the previous results to the case when d.f .x 1 ; : : : ; x n //f .x 1 ; : : : ; x n / f .x 1 ; : : : ; x n /ı.f .x 1 ; : : : ; x n // 2 Z.R/ for all x 1 ; : : : ; x n in a non-zero ideal of R, where f .x 1 ; : : : ; x n / is a non-central polynomial over the ring K, and showed that either d D ı D 0 or d D ı, while the polynomial f .x 1 ; : : : ; x n / 2 is central-valued on R, except when char.R/ D 2 and R satisfies s 4 . Recently in [24], Niu and Wu studied the left annihilator of the set ¹d.u/u uı.u/ W u 2 Lº, where d and ı are derivations of R and L is a noncentral Lie ideal of R. In case the annihilator is not zero, the conclusion is that R satisfies the standard identity s 4 and d D ı are inner derivations.…”

Section: Introductionmentioning

confidence: 81%

Annihilating conditions on generalized derivations acting on multilinear polynomials

Filippis¹,

Dhara²,

Scudo³

2013

Georgian Mathematical Journal

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Let K be a commutative ring with unity, R be a non-commutative prime K-algebra of characteristic different from 2, U be the Utumi quotient ring of R, C be the extended centroid of R, and G and H be two non-zero generalized derivations of R. Suppose that f .x 1 ; : : : ; x n / is a non-central multilinear polynomial over K and there exists 0 ¤ a 2 R such that a.G.f .X//f .X / f .X/H.f .X /// D 0 for all X D .x 1 ; : : : ; x n / 2 R n . Then one of the following holds: (1) there exist b 0 ; c 0 2 U such that G.x/ D b 0 x C xc 0 and H.x/ D c 0 x for all x 2 R, with ab 0 D 0;(2) f .x 1 ; : : : ; x n / 2 is central-valued on R and there exist b 0 ; c 0 ; q 0 2 U such that G.x/ D b 0 x Cxc 0 and H.x/ D c 0 x Cxq 0 for all x 2 R, with a.b 0 q 0 / D 0.

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“…In [13] Wong extended the previous result to the case when the elements of the Lie ideal are replaced by all the evaluations of a multilinear polynomial. Finally in [10] Lee and Shiue generalize the theorem of Wong for any polynomial (without any assumption on multilinearity). More precisely they show that if d, g are derivations of R and f (x 1 , .…”

Section: Introductionmentioning

confidence: 93%