On ($\sigma$-$\delta$)-rings over Noetherian rings

For a ring $R$, an endomorphism $\sigma$ of $R$ and a $\sigma$-derivation $\delta$ of $R$, we introduce ($\sigma$-$\delta$)-ring and ($\sigma$-$\delta$)-rigid ring which are the generalizations of $\sigma(*)$-rings and $\delta$-rings, and investigate their properties. Moreover, we prove that a ($\sigma$-$\delta$)-ring is $2$-primal and its prime radical is completely semiprime.