Summary: Let \(R\) be a ring with an automorphism \(\sigma\). An ideal \(I\) of \(R\) is a `\(\sigma\)-ideal' of \(R\) if \(\sigma(I)=I\). A proper ideal \(P\) of \(R\) is a `\(\sigma\)-prime ideal' of \(R\) if \(P\) is a \(\sigma\)-ideal of \(R\) and for \(\sigma\)-ideals \(I\) and \(J\) of \(R\), \(IJ\subseteq P\) implies that \(I\subseteq P\) or \(J\subseteq P\). A proper ideal \(Q\) of \(R\) is a `\(\sigma\)-semiprime ideal' of \(R\) if \(Q\) is a \(\sigma\)-ideal and for a \(\sigma\)-ideal \(I\) of \(R\), \(I^2\subseteq Q\) implies that \(I\subseteq Q\). The \(\sigma\)-prime radical is defined by the intersection of all \(\sigma\)-prime ideals of \(R\) and is denoted by \(P_\sigma(R)\). In this paper, the following results are obtained: (1)~For a principal ideal domain \(R\), \(P_\sigma(R)\) is the smallest \(\sigma\)-semiprime ideal of \(R\); (2)~For any ring \(R\) with an automorphism \(\sigma\) and for a skew Laurent polynomial ring \(R[x,x^{-1};\sigma]\), the prime radical of \(R[x,x^{-1};\sigma]\) is equal to \(P_\sigma(R)[x,x^{-1};\sigma]\).