Maximality of orders in Dedekind domains. II (Q2043743)

Let \(R\) be a Dedekind domain and let \(q(R)\) be its field of fractions. A subring \(\mathcal{O}\) of \(R\) (containing the unit of \(R\)) is said to be an order in \(R\) if \(R\) is a finitely-generated \(\mathcal{O}\)-module and \(q(R)\) equals the field of fractions \(q(\mathcal{O})\) of \(\mathcal{O}\). When \(\mathcal{O}\) is an order in \(R\), it is a Noetherian domain of dimension \(1\) and \(R\) is its integral closure in \(q(R)\); such being the case, \(\mathfrak{f}\) denotes the conductor of \(\mathcal{O}\), i.e., the greatest ideal of \(R\) included in \(\mathcal{O}\). Henceforth, we assume that \(\mathcal{O}\) is an order in \(R\). Then \(C(\mathcal{O})\) denotes the group of Cartier divisors of \(\mathcal{O}\), i.e., the multiplicative group generated by all invertible ideals in \(\mathcal{O}\), and \(\mathrm{Pic}(\mathcal{O})\) stands for the Picard group of \(\mathcal{O}\), i.e., the additively written group of Cartier divisors of \(\mathcal{O}\) modulo the principal ideals \(a\mathcal{O}\), \(0 \neq a \in \mathcal{O}\). Also, \(\mathrm{Div}(\mathcal{O})\) is the group of Weil divisors of \(\mathcal{O}\), i.e., the free abelian group generated by all maximal ideals in \(\mathcal{O}\). The natural homomorphism \(C(\mathcal{O}) \to C(R)\), defined by \(J \to JR\), for all \(J \in C(\mathcal{O})\) is called the Cartier group homomorphism; the surjective natural homomorphism \(\mathrm{Pic}(\mathcal{O}) \to \mathrm{Pic}(R)\) mapping the class of \(J\) in \(C(\mathcal{O})\) into the class of \(JR\) in \(\mathrm{Pic}(R)\), for every \(J \in C(\mathcal{O})\) is called the Picard group homomorphism. \par The paper under review is a continuation of Part I [the author, J. Algebra Appl. 19, No. 7, Article ID 2050125, 13 p. (2020; Zbl 1460.13029)], which provides new conditions equivalent to the one that \(\mathcal{O} = R\). The main results of the paper under review, in contrast to the quoted one, are obtained without the assumption that \(\mathrm{Pic}(R)\) is a torsion group. They show the equivalence of a number of conditions including the following: \begin{itemize} \item[(a)] \(\mathcal{O} = R\); \item[(b)] the Cartier group homomorphism is an isomorphism; \item[(c)] the Picard group homorphism is an isomorphism and \(U(R) \subseteq \mathcal{O}\); \item[(d)] \(C(\mathcal{O})\) is a torsion-free group; \item[(e)] The ring extension \(R/\mathcal{O}\) is radical, i.e., for each \(r \in R\), there exists \(s \in \mathbb{N}\), such that \(r^s \in \mathcal{O}\); \item[(f)] the length homomorphism \(C(\mathcal{O}) \to \mathrm{Div}(\mathcal{O})\) is an isomorphism. \end{itemize} One of them states that \(\mathcal{O} = R\) if and only if all of the following conditions hold: \begin{itemize} \item[(i)] the natural homomorphism \(W\mathcal{O} \to Wq(\mathcal{O})\) is injective; \item[(ii)] \(R/\mathcal{O}\) is a radical extension; \item[(iii)] \(C(\mathcal{O})\) does not contain a nontrivial element of odd order; \item[(iv)] \(U(R) \subseteq \mathcal{O}\). \end{itemize} The author gives an example showing that the fulfillment of conditions (ii), (iii) and (iv) does not guarantee that \(\mathcal{O} = R\).