PRIME RADICALS OF SKEW LAURENT POLYNOMIAL RINGS

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 $^{-1}$ ; $\sigma$ ].