Derivations modulo elementary operators

JOURNAL OF ALGEBRA

卷:338

Article

Chuang, Chen-Lian¹  Lee, Tsiu-Kwen¹ 

[1] Natl Taiwan Univ, Dept Math, Taipei 106, Taiwan

关键词: Derivation;    Idempotent;    Prime ring;    Elementary operator;    Zero-product preserving;    Functional identity;   

DOI  :  10.1016/j.jalgebra.2011.05.009

来源: Elsevier

【 摘 要 】

Let R be a prime ring with extended centroid C and symmetric Martindale quotient ring Q(s)(R). Suppose that Q(s)(R) contains a nontrivial idempotent e such that eR + Re subset of R. Let phi : R x R -> RC + C be the bi-additive map (x, y) -> G(x)y + xH(y) + Sigma(i)a(i)xb(i)yc(i), where G, H : R -> R are additive maps and where a(i), b(i), c(i) is an element of RC + C are fixed. Suppose that phi is zero-product preserving, that is, phi(x, y) = 0 for x, y is an element of R with xy = 0. Then there exists a derivation delta : R -> Q(s)(RC) such that both G and H are equal to delta plus elementary operators. Moreover, there is an additive map F : R -> Q(s)(RC) such that phi(x, y) = F(xy) for all x, y is an element of R. The result is a natural generalization of several related theorems in the literature. Actually we prove some more general theorems. (C) 2011 Elsevier Inc. All rights reserved.

【 授权许可】

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