(untitled) (European Journal of Pure and Applied Mathematics, 2013)
Let R be a ring and Ïƒ an endomorphism of R. Recall that R is said to be a Ïƒ(âˆ—)-ring if aÏƒ(a) âˆˆ P(R) implies a âˆˆ P(R) for a âˆˆ R, where P(R) is the prime radical of R. We also recall that R is said to be a weak Ïƒ-rigid ring if aÏƒ(a) âˆˆ N(R) if and only if a âˆˆ N(R) for a âˆˆ R, where N(R) is the set of nilpotent elements of R. In this paper we give a relation between a Ïƒ(âˆ—)-ring and a weak Ïƒ-rigid ring. We also give a necessary and sufficient condition for a Noetherian ring to be a weak Ïƒ-rigid ring. Let Ïƒ be an endomorphism of a ring R. Then Ïƒ can be extended to an endomorphism (say Ïƒ) of R[x;Ïƒ]. With this we show that if R is a Noetherian ring and Ïƒ an automorphism of R, then R is a weak Ïƒ-rigid ring if and only if R[x;Ïƒ] is a weak Ïƒ-rigid ring.Â