On Additive Mappings in a ∗-Ring with an Identity Element
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Let R be a semiprime ring with involution ∗ and let F, D : R → R be additive mappings satisfying the conditions (i) F(x 2) = F(x)x ∗+x ∗ D(x) and D(x 2) = D(x)x ∗+x ∗ D(x); (ii) F(x n + 1) = F(x)(x ∗) n +x ∗ D(x)(x ∗) n − 1+(x ∗)2 D(x)(x ∗) n − 2+⋯+(x ∗) n D(x) for all x ∈ R. Then, F(x y) = F(y)x ∗+y ∗ D(x) and D(x y) = D(y)x ∗+y ∗ D(x) for all x, y ∈ R.
KeywordsAdditive mappings Semiprime rings and involution
Mathematics Subject Classification (2010)16W25 16N60 16R50
The authors are greatly indebted to the referee for his/her several useful suggestions and valuable comments to improve the presentation of this paper.