Ore Extensions over Weak -rigid Rings and (∗)-rings

Let R be a ring andan endomorphism of a ring R. Recall that R is said to be a �(∗)-ring if a�(a)∈ P(R) implies a∈ P(R) for a∈ R, where P(R) is the prime radical of R. We also recall that R is said to be a weak �-rigid ring if a�(a)∈ N(R) if and only if a∈ N(R) for a∈ R, where N(R) is the set of nilpotent elements of R. In this paper we give a relation between a �(∗)-ring and a weak �-rigid ring. We also give a necessary and sufficient condition for a Noetherian ring to be a weak �-rigid ring. Letbe an endomorphism of a ring R anda �-derivation of R such that �(�(a)) = �(�(a)) for all a∈ R. Thencan be extended to an endomorphism (say �) of R(x;�,�) andcan be extended to a �-derivation (say �) of R(x;�,�). With this we show that if R is a 2-primal commutative Noetherian ring which is also an algebra overQ (whereQ is the field of rational numbers), � is an automorphism of R anda �-derivation of R such that �(�(a))= �(�(a)) for all a∈ R, then R is a weak �-rigid ring implies that R(x;�,�) is a weak �-rigid ring. 2000 Mathematics Subject Classifications: 16S36, 16P40, 16P50, 16U20,16W25