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- W2583152169 abstract "Let φ(x)=xp be the Frobenius map on an associative unital ring R with prime characteristic p. It is well-known that, whenever R is commutative, φn is a ring homomorphism, for all positive integers n. The converse, however, is not true in general. Indeed, we prove that, if R is m-Engel, for some positive integer m, then there exists a positive integer n0 depending only on m such that, for all n≥n0, φn is a ring homomorphism with central image. Conversely, if any one of the following conditions holds: φn respects addition, φn respects multiplication, φn respects Lie multiplication, or the image of φn is commutative, then R is m-Engel, for some m depending only on n. Consequently, if φ is surjective, and any one of the aforementioned conditions holds, then R must be commutative." @default.
- W2583152169 created "2017-02-10" @default.
- W2583152169 creator A5067690564 @default.
- W2583152169 date "2017-06-01" @default.
- W2583152169 modified "2023-09-27" @default.
- W2583152169 title "When is a power of the Frobenius map on a noncommutative ring a homomorphism?" @default.
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- W2583152169 doi "https://doi.org/10.1016/j.jalgebra.2017.01.026" @default.
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