Additive mappings on semiprime rings functioning as centralizers. (English) Zbl 1513.16018
Summary: The objective of this research is to prove that an additive mapping \(\mathcal{T}:R \rightarrow R\) is a centralizer on \(R\) if it satisfies any one of the following identities:
- (i)
- \(3\mathcal{T}(x^{3n})=\mathcal{T}(x^n)x^{2n}+x^n\mathcal{T}(x^n)x^n+x^{2n}\mathcal{T}(x^n)\)
- (ii)
- \(2\mathcal{T}(x^{2n})=\mathcal{T}(x^n)x^n+x^n\mathcal{T}(x^n)\)
- (iii)
- \(\mathcal{T}(x^{3n})=x^n\mathcal{T}(x^n)x^n\)
MSC:
| 16N60 | Prime and semiprime associative rings |
| 16W25 | Derivations, actions of Lie algebras |
| 16R50 | Other kinds of identities (generalized polynomial, rational, involution) |
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