Bulletin of the
Korean Mathematical Society BKMS

We show that the $\theta$-prime radical of a ring $R$ is the set of all strongly $\theta$-nilpotent elements in $R$, where $\theta$ is an automorphism of $R$. We observe some conditions under which the $\theta$-prime radical of $R$ coincides with the prime radical of $R$. Moreover we characterize elements in prime radicals of skew Laurent polynomial rings, studying $(\theta,\theta^{-1})$-(semi)primeness of ideals of $R$.

Keywords: $\theta$-ideal, $\theta$-prime ideal, $\theta$-semiprime ideal, strongly $\theta$-nilpotent element, $\theta$-prime radical, prime radical, skew polynomial ring, skew Laurent polynomial ring

MSC numbers: 16N40, 16N60, 16S36