On derivations and commutativity in prime rings

Vincenzo De Filippis

doi:10.1155/s0161171204403536

Vincenzo De Filippis

Open Access

https://doi.org/10.1155/s0161171204403536

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Abstract

Let R be a prime ring of characteristic different from 2, d a nonzero derivation of R, and I a nonzero right ideal of R such that [[d(x), x], [d(y), y]] = 0, for all x, y ∈ I. We prove that if [I, I]I ≠ 0, then d(I)I = 0.

Highlights

Let R be a prime ring and d a nonzero derivation of R
Let R be a prime ring of characteristic different from 2, d a nonzero derivation of R, and I a nonzero right ideal of R such that [[d(x), x], [d(y), y]] = 0, for all x, y ∈ I
We fix our attention on the Engel condition [[d(x1), x1], x2]

Summary

Introduction

Let R be a prime ring and d a nonzero derivation of R. Let R be a prime ring of characteristic different from 2, d a nonzero derivation of R and I a nonzero right ideal of R such that [[d(x), x], [d(y), y]] = 0, for all x, y ∈ I. I, IR, and IQ satisfy the same generalized polynomial identities with coefficients in Q.

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