Intermediary rings in normal pairs
All rings considered are integral domains. A pair of rings \((R,S)\) is called a \textit{normal pair} if \(R\subseteq S\) and every ring contained between \(R\) and \(S\) is integrally closed in \(S\). Given a ring extension \(R\subset S\), the symbol \([R,S]\) denotes the set of all rings between \(R\) and \(S\), including the end-points. The authors are interested in the cardinality of the latter interval of rings. Notation needed for the main result is as follows: \(\text{Supp}(S/R)=\{ P\in \text{Spec}(R) \mid PS=S\}\) (as an ordered set) and it is assumed to contain \(n\) maximal elements. For a non-zero ideal \(I\) of \(R\), denote \(\Omega_R(I)=\{ x\in qf(R) \mid \forall y\in I\exists n\geq 1\,\, xy^n\in R\}\) where qf stands for the quotient field; define inductively \(\Gamma_1=\{\{Q\} \mid Q\in \text{Supp}(S/R)\}\) and for \(2\leq k\leq n\) denote by \(\Gamma_k\) the set of all subsets \(\{Q_1,\dots,Q_k\}\) of \(\text{Supp}(S/R)\) made up of \(k\) pairwise incompatible elements. The main results is: Theorem 2.4. Every \(T\in [R,S]\setminus \{R\}\) is uniquely represented as \(T=\Omega_R(Q_1Q_2\dots Q_k)\), where \(\{Q_1,Q_2,\dots Q_k\}\in \bigcup_{i=1}^n\Gamma_i\). Consequently, the cardinality of \([R,S]\) is \(1+| \text{Supp}(S/R)| + \sum_{i=2}^n| \Gamma_i| \). Corollary 2.5. \(| \text{Supp}(S/R)| +2^n-n\leq | [R,S]| \leq 2^{| \text{Supp}(S/R)| }\). Corollary 2.7. \([R,S]\) is totally ordered iff its cardinality is by 1 greater than the cardinality of \(\text{Supp}(S/R)\), iff \(R\) has a unique maximal overring in \([R,S]\), iff \(\text{Supp}(S/R)\) is totally ordered. Theorem 3.1 \(\text{Supp}(S/R)\) is contained in the set of maximal ideals of \(R\) iff the cardinality of \([R,S]\) is \(2^{| \text{Supp}(S/R)| }\). Corollary 3.2. If \(R\) is noetherian, then the cardinality of \([R,S]\) is \(2^m\), where \(m\) is the set of maximal ideals \(M\) of \(R\) with \(\text{ht}(M)=1\) and \(MS=S\).
Kaplansky transform of a ring with respect to an ideal
scientific article; zbMATH DE number 5299435
scientific article