Let R be a ring with an automorphism 17. An ideal [ of R is ($sigma$-ideal of R if $sigma$(I).= I. A proper ideal P of R is ($sigma$-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 $sigma$-semiprime ideal of Q 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$_{(R). In this paper, the following results are obtained: (1) For a principal ideal domain R, P$_{(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$_{(R)[x, x$^{-1}$; $sigma$ ].