Skew-Laurent rings over () -rings

Let R be an associative ring with identity 16= 0, and an endomorphism of R. We recall () property on R (i.e. a (a)2 P(R) implies a2 P(R) for a2 R, where P(R) is the prime radical of R). Also recall that a ring R is said to be 2-primal if and only if P(R) and the set of nilpotent elements of R coincide, if and only if the prime radical is a completely semiprime ideal. It can be seen that a () -ring is a 2-primal ring. Let R be a ring and an automorphism of R. Then we know that can be extended to an automorphism (say ) of the skew-Laurent ring R(x, x 1 ; ). In this paper we show that if R is a Noetherian ring and is an automorphism of R such that R is a () -ring, then R(x, x 1 ; ) is a () -ring. We also prove a similar result for the general Ore extension R(x; , ), where is an automorphism of R and a -derivation of R. 2010 Mathematics Subject Classifications: 16-XX; 16N40, 16P40, 16S36.