Abstract

Let R be a commutative ring and X^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 , 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 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 Ti-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.

摘要

令R为交换环，X ^i.c ( R ) 表示R的所有积分闭最大子环的集合。证明如果R是非域G域，则存在S ∈ X ^i.c ( R ) 且 ( S : R ) = 0。如果K是非绝对代数的代数闭域，则证明多项式环K [ X ] 也有一个具有零导体的整体闭合最大子环；K的整体闭合最大子环的表征[给出了具有（非）零导体的X ]。可以观察到，K [ X ] 的一个整体闭合的极大子环S是一个主理想域 (PID) 当且仅当M = Sq对于一些q ^{− 1} ∈ K \ S，其中M是扩展S ⊆ K [ X ]。我们证明如果f ( X, Y ) 是K [ X, Y ] 中的一个不可约多项式，则存在K [ X, Y ] 与 ( S : K [ X, Y ]) = f ( X, Y ) K [ X, Y ]。证明了，如果R是一个环并且，其中I是R的理想，则是R的理想} 是X ( R ) 上的闭集拓扑。我们证明了这个空间具有类似的属性，例如Spec ( R ) 或K ⁿ （affine 空间）上的 Zariski 空间中的那些。特别是，如果K是一个在其素数子环上不是代数的域，则X ^i.c ( K [ X ₁ ,…, X _n ]) 是不可约的，如果另外K是代数闭的，那么我们证明了 Hilbert Nullstellensatz 的类似完整形式K [ X ₁ ,…, X _n ]。此外，如果R是非域G域或R = K [ X ]，其中K是代数闭域，在其素数子环上不是代数域，则 ∅ ≠ gen(X ^i.c (R )) = { S ∈ X ^i.c ( R ) | ( S : R )=0}。我们准确地确定空间X ^i.c ( R ) 何时是i = 0 , 1 , 2 的Ti空间。特别是，我们证明如果X ^i.c是T 1-空间，则R是希尔伯特环和|X ^i.c ( R )| ≤ 2 ^{| 最大（R）|} . 最后，我们确定当空间X ^i.c已连接。