Abstract
The note concerns the commutativity of associative rings (possibly nonunital) endowed with a derivation. Our focus is on ?-prime rings. We give a new proof of Hirano and Tominaga?s result that a ?-prime ring is commutative whenever the derivation ? is nonzero and commuting on a nonzero two-sided ?-ideal. We also provide some further generalizations of Herstein?s classical theorem on a prime ring admitting a nonzero derivation with commutative range.
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