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

Abstract

Let R be a commutative ring and R^i.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 ∈ X^i.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 S (I) = {T ∈ X^i:c(R)  | I ⊆ T}, where I is an ideal of R, then S :=  {S(I) | I is an ideal of R} is a topology for closed sets on X^i:c(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 X^i.c (K [X₁...,X_n]) is irreducible and if in addition K is algebraically closed, then we prove a similar full form of the Hilbert Nullstellensatz for K[X₁,...,X_n]. 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 ∈ X^i.c(R) | (S : R) = 0}. We determine exactly when the space X^i.c(R) is a T_i - space for i = 0, 1,2. In particular, we show that if X^i:c is T₁-space then R is a Hilbert ring and |X^i:c(R)| ≤ 2^|Max(R)|. Finally, we determine when the space X^i.c is connected.