Hilbert basis theorem for rings with *-noetherian spectrum

Let R be a commutative ring with identity, R[X] the polyno-mial ring over R, ∗ a radical operation on R and ⋆ a radical operation of finite character on R[X]. In this paper, we give Hilbert basis theorem for rings with ∗-Noetherian spectrum. More precisely, we show that if (I^∗R[X])^⋆ = (IR[X])^⋆ and (I^∗R[X])^⋆ ∩ R = I^∗ for all ideals I of R, then R has ∗-Noetherian spectrum if and only if R[X] has ⋆-Noetherian spec-trum. This is a generalization of a well-known fact that R has Noetherian spectrum if and only if R[X] has Noetherian spectrum.