(untitled) (European Journal of Pure and Applied Mathematics, 2010)
Let R be a ring and σ an endomorphism of a ring R. Recall that R is said to be a σ(*)-ring if aσ(a)P(R) implies aP(R) for aR, where P(R) is the prime radical of R. We also recall that R is said to be a weak σ-rigid ring if aσ(a)N(R) if and only if aN(R) for aR, where N(R) is the set of nilpotent elements of R. In this paper we give a relation between a σ(*)-ring and a weak σ-rigid ring. We also give a necessary and sufficient condition for a Noetherian ring to be a weak σ-rigid ring. Let σ be an endomorphism of a ring R and δ a σ-derivation of R such that σ(δ(a)) = δ(σ(a)) for all aR. Then σ can be extended to an endomorphism (say σ) of R[x;σ,δ] and δ can be extended to a σ-derivation (say δ) of R[x;σ,δ]. With this we show that if R is a 2-primal commutative Noetherian ring which is also an algebra over Q (where Q is the field of rational numbers), σ is an automorphism of R and δ a σ-derivation of R such that σ(δ(a)) = δ(σ(a)) for all aR, then R is a weak σ-rigid ring implies that R[x;σ,δ] is a weak σ-rigid ring.