PRIME RADICALS IN ORE EXTENSIONS

Let R be a ring with an endomorphism and a derivation . An ideal I of R is ()-ideal of R if and . An ideal P of R is a ()-prime ideal of R if P() is a ()-ideal and for ()-ideals I and J of R, implies that or . An ideal Q of R is ()-semiprime ideal of R if Q is a ()-ideal and for ()-ideal I of R, implies that . The ()-prime radical (resp. prime radical) is defined by the intersection of all ()-prime ideals (resp. prime ideals) of R and is denoted by (resp. P(R)). In this paper, the following results are obtained: (1) is the smallest ()-semiprime ideal of R; (2) For every extended endomorphism of , the -prime radical of an Ore extension is equal to .