The dimension-overrings equation and maximal ideals of integral domains (Q6547693)

Let \(R\) be an integral domain with quotient field \(K\) and \(O(R)\) the set of all overrings of \(R\). In this paper, the author investigated integral domains with only finitely many overrings and establish several new sharp inequalities relating the cardinality of the set \(O(R)\), the Krull dimension, and the number of maximal ideals. A first main result states that for a Prüfer domain \(R\) with finite Krull dimension, \(\dim(R)\), and finite number \(m\) of maximal ideals, \N\[\N(2^{m} + \dim(R)-1)\leq |O(R)|\leq (1 + \dim(R))^{m}\N\]\NA second main result states that for an integrally closed domain \(R\) with finite Krull dimension \(d\geq 1\), and a positive integer \(n\), if \(|O(R)| = n + d\), then the following statements hold true: \((1)\) The number of maximal ideals is finite and satisfies the inequalities: \(\log_{d+1}(n + d) \leq |\mathrm{Max}(R)|\leq \log_{2}(n + 1)\).\N\N\((2)\) \(R\) has exactly \(m = \log_{d+1}(n+d)\) maximal ideals if and only if \(n = (d+1)m-d\) and \(R\) is a Prüfer domain with spectrum isomorphic to a tree consisting of \(m\) chains each containing \(d + 1\) elements and meeting only at their respective minimal elements.\N\N\((3)\) \(R\) has exactly \(m = \log_{2}(n +1)\) maximal ideals if and only if \(n = 2m-1\) and \(R\) is a Prüfer domain with spectrum isomorphic to a tree consisting of a chain containing \(d\) elements and connected at the top element with \(m\) additional edges to each of the \(m\) maximal ideals. Other inequalities related to length, dimension and number of maximal ideals are given.