Prime Radicals of Skew Laurent Polynomial Rings

Abstract

Let R be a ring withan automorphism sigma. An ideal I of R is sigma-idealof R if sigma(I) = I. A proper ideal P of R is{itsigma-prime ideal/} of R if P is a sigma-ideal of Rand for sigma-ideals I and J of R, IJ subseteq Pimplies that I subseteq P or J subseteq P. A properideal Q of R is {itsigma-semiprime ideal/} of Q if Q isa sigma-ideal and for a sigma-ideal I of R, I^{2}subseteq Q implies that I subseteq Q. Thesigma-prime radical is defined by the intersection of allsigma-prime ideals of R and is denoted by P_{sigma}(R).In this paper, the following results are obtained: (1)For aprincipal ideal domain R, P_{sigma}(R) is the smallestsigma-semiprime ideal of R; (2)For any ring R with anautomorphism sigma and for a skew Laurent polynomial ring R[x,x^{-1}; sigma], the prime radical of R[x, x^{-1}; sigma] isequal to P_{sigma}(R)[x, x^{-1}; sigma].