On Two Commutativity Criteria for $\boldsymbol\delta$-Prime Rings

Kamil Kular, Marcin Skrzyński

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