Nonadditive strong commutativity preserving derivations and endomorphisms
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Let S be a nonempty subset of a ring R. A map f : R → R is called strong commutativity preserving on S if [f(x), f(y)] = [x, y] for all x, y ∈ S, where the symbol [x, y] denotes xy − yx. Bell and Daif proved that if a derivation D of a semiprime ring R is strong commutativity preserving on a nonzero right ideal of R, then p⊆ Z, the center of R. Also they proved that if an endomorphism T of a semiprime ring R is strong commutativity preserving on a nonzero two-sided ideal I of R and not identity on the ideal I ∪ T−1(I), then R contains a nonzero central ideal. This short note shows that the conclusions of Bell and Daif are also true without the additivity of the derivation D and the endomorphism T.
Abstract
Let S be a nonempty subset of a ring R. A map f : R → R is called strong commutativity preserving on S if [f(x), f(y)] = [x, y] for all x, y ∈ S, where the symbol [x, y] denotes xy − yx. Bell and Daif proved that if a derivation D of a semiprime ring R is strong commutativity preserving on a nonzero right ideal of R, then p⊆ Z, the center of R. Also they proved that if an endomorphism T of a semiprime ring R is strong commutativity preserving on a nonzero two-sided ideal I of R and not identity on the ideal I ∪ T−1(I), then R contains a nonzero central ideal. This short note shows that the conclusions of Bell and Daif are also true without the additivity of the derivation D and the endomorphism T.
참고문헌 (14)
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