Characterization of Additive Maps on Rings Behaving Like Derivations at Idempotent-Product Elements

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Heidari, Hananeh | 2015

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Type of Document: M.Sc. Thesis
Language: Farsi
Document No: 46803 (02)
University: Sharif University of Technology
Department: Mathematical Sciences
Advisor(s): Mahdavi Hezavehi, Mohammad
Abstract:
Defining the structure of maps using local features is among the popular fields of study in mathematics. Therefore determining the structure of maps on rings which behave like derivations at idempotent-product elements has been getting attention recently. This subject is useful for examining the structure of rings and algebraic operators in both algebra and analysis as well. Suppose that R is a ring, d : R ! R is an additive map, z 2 R and d meets the condition below: 8a; b 2 R : d(ab) = ad(b) + d(a)b Therefore d is called a derivation on R. If for every a; b 2 R where ab = z, d(ab) = ad(b) + d(a)b then d behaves like a derivation at idempotent-product elements of ab = z. The main challenge is defining the structure of d and answering whether d is a derivation or not. In the present thesis, we assume that R is a ring-with-identity and has an implicit idempotent p (p ̸= 0; 1; p2 = p): We are going to answer the above question in the case of z = 0, z = 1 or z = p and in the end present applications for some of the algebraic operators
Keywords:
Banach Spaces ; Derivatives ; Idempotent Rings ; Algebra ; Additive Maps