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