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Derivations modulo elementary operators

Authors
Journal
Journal of Algebra
0021-8693
Publisher
Elsevier
Publication Date
Volume
338
Issue
1
Identifiers
DOI: 10.1016/j.jalgebra.2011.05.009
Keywords
  • Derivation
  • Idempotent
  • Prime Ring
  • Elementary Operator
  • Zero-Product Preserving
  • Functional Identity
Disciplines
  • Mathematics

Abstract

Abstract 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 e R + R e ⊆ R . Let ϕ : R × R → R C + C be the bi-additive map ( x , y ) ↦ G ( x ) y + x H ( y ) + ∑ i a i x b i y c i , where G , H : R → R are additive maps and where a i , b i , c i ∈ R C + C are fixed. Suppose that ϕ is zero-product preserving, that is, ϕ ( x , y ) = 0 for x , y ∈ R with x y = 0 . Then there exists a derivation δ : R → Q s ( R C ) such that both G and H are equal to δ plus elementary operators. Moreover, there is an additive map F : R → Q s ( R C ) such that ϕ ( x , y ) = F ( x y ) for all x , y ∈ R . The result is a natural generalization of several related theorems in the literature. Actually we prove some more general theorems.

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