A Zariski topology on integrally closed maximal subrings of a commutative ring

Abstract

Let R be a commutative ring and Xi.c (R) denotes the set of all integrally closed maximal subrings of R. It is shown that if R is a non-field G-domain, then there exists S ∈ Xi.c (R) with (S : R) = 0. If K is an algebraically closed field which is not absolutely algebraic, then we prove that the polynomial ring K[X] has an integrally closed maximal subring with zero conductor too; a characterization of integrally closed maximal subrings of K[X] with (non-)zero conductor is given. It is observed that, an integrally closed maximal subrings S of K[X] is a principal ideal domain (PID) if and only if M = Sq for some q− 1 ∈ K \\ S, where M is the crucial maximal idea of the extension S ⊆ K[X]. We show that if f (X, Y) is an irreducible polynomial in K[X, Y ], then there exists an integrally closed maximal subring S of K[X, Y ] with (S : K[X, Y ]) = f (X, Y) K[X, Y ]. It is proved that, if R is a ring and , where I is an ideal of R, then is an ideal of R} is a topology for closed sets on X (R). We show that this space has similar properties such as those one in the Zariski spaces on Spec(R) or Kn (the affine space). In particular, if K is a field which is not algebraic over its prime subring, then Xi.c (K[X 1,…, Xn ]) is irreducible and if in addition K is algebraically closed, then we prove a similar full form of the Hilbert Nullstellensatz for K[X 1,…, Xn ]. Moreover, if R is a non-field G-domain or R = K[X], where K is an algebraically closed field which is not algebraic over its prime subring, then ∅ ≠ gen(X i.c (R)) = {S ∈ Xi.c (R) | (S : R) = 0}. We determine exactly when the space Xi.c (R) is a Ti-space for i = 0, 1, 2. In particular, we show that if Xi.c is T1-space then R is a Hilbert ring and |Xi.c (R)| ≤ 2|Max(R)|. Finally, we determine when the space Xi.c is connected.