Derivations cocentralizing multilinear polynomials on left ideals

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Abstract

Let R be a prime ring with extended centroid C, λ a nonzero left ideal of R and f(X1, . . ., Xt) a nonzero multilinear polynomial over C. Suppose that d and δ are derivations of R such that d(f(x1,...,xt))f(x1,...,xt)-f(x1,...,xt)δ(f(x1,...,xt))ε C for all x1,...,xt ε λ. Then either d = 0 and λδ(λ) = 0 or λC = RCe for some idempotent e in the socle of RC and one of the following holds:(1) f(X1, . . ., Xt) is central-valued on eRCe;(2) λ(d + δ)(λ) = 0 and f (X1, . . ., Xt)2 is central-valued on eRCe;(3) char R = 2 and eRCe satisfies st4(X1, X2, X3, X4), the standard polynomial identity of depsiee 4.

Original languageEnglish
Pages (from-to)297-311
Number of pages15
JournalMonatshefte fur Mathematik
Volume162
Issue number3
DOIs
Publication statusPublished - 2011 Mar 1

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Extended Centroid
Socle
Polynomial Identities
Prime Ring
Idempotent
Polynomial
Standards

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Cite this

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abstract = "Let R be a prime ring with extended centroid C, λ a nonzero left ideal of R and f(X1, . . ., Xt) a nonzero multilinear polynomial over C. Suppose that d and δ are derivations of R such that d(f(x1,...,xt))f(x1,...,xt)-f(x1,...,xt)δ(f(x1,...,xt))ε C for all x1,...,xt ε λ. Then either d = 0 and λδ(λ) = 0 or λC = RCe for some idempotent e in the socle of RC and one of the following holds:(1) f(X1, . . ., Xt) is central-valued on eRCe;(2) λ(d + δ)(λ) = 0 and f (X1, . . ., Xt)2 is central-valued on eRCe;(3) char R = 2 and eRCe satisfies st4(X1, X2, X3, X4), the standard polynomial identity of depsiee 4.",
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Derivations cocentralizing multilinear polynomials on left ideals. / Liu, Cheng-Kai.

In: Monatshefte fur Mathematik, Vol. 162, No. 3, 01.03.2011, p. 297-311.

Research output: Contribution to journalArticle

TY - JOUR

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AU - Liu, Cheng-Kai

PY - 2011/3/1

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N2 - Let R be a prime ring with extended centroid C, λ a nonzero left ideal of R and f(X1, . . ., Xt) a nonzero multilinear polynomial over C. Suppose that d and δ are derivations of R such that d(f(x1,...,xt))f(x1,...,xt)-f(x1,...,xt)δ(f(x1,...,xt))ε C for all x1,...,xt ε λ. Then either d = 0 and λδ(λ) = 0 or λC = RCe for some idempotent e in the socle of RC and one of the following holds:(1) f(X1, . . ., Xt) is central-valued on eRCe;(2) λ(d + δ)(λ) = 0 and f (X1, . . ., Xt)2 is central-valued on eRCe;(3) char R = 2 and eRCe satisfies st4(X1, X2, X3, X4), the standard polynomial identity of depsiee 4.

AB - Let R be a prime ring with extended centroid C, λ a nonzero left ideal of R and f(X1, . . ., Xt) a nonzero multilinear polynomial over C. Suppose that d and δ are derivations of R such that d(f(x1,...,xt))f(x1,...,xt)-f(x1,...,xt)δ(f(x1,...,xt))ε C for all x1,...,xt ε λ. Then either d = 0 and λδ(λ) = 0 or λC = RCe for some idempotent e in the socle of RC and one of the following holds:(1) f(X1, . . ., Xt) is central-valued on eRCe;(2) λ(d + δ)(λ) = 0 and f (X1, . . ., Xt)2 is central-valued on eRCe;(3) char R = 2 and eRCe satisfies st4(X1, X2, X3, X4), the standard polynomial identity of depsiee 4.

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