On weak (σ, δ)-rigid rings over Noetherian rings

Let R be a Noetherian integral domain which is also an algebra over Q (Q is the field of rational numbers). Let sigma be an endomorphism of R and delta a sigma-derivation of R. We recall that a ring R is a weak (sigma, delta)-rigid ring if a(sigma(a)) + delta(a)) is an element of N(R) if and only if a is an element of N(R) for a is an element of R (N(R) is the set of nilpotent elements of R). With this we prove that if R is a Noetherian integral domain which is also an algebra over Q, sigma an automorphism of R and delta a sigma-derivation of R such that R is a weak (sigma, delta)-rigid ring, then N(R) is completely semiprime.