Generalized Dedekind domains and their injective modules (Q1328770)

In this paper the author studies structural properties of generalized Dedekind domains and their injective modules. A domain \(R\) is a generalized Dedekind domain iff it is a Prüfer domain, \(P \neq P^2\) for every nonzero prime ideal \(P\) of \(R\), and every prime ideal of \(R\) is the radical of a finitely generated ideal. The principal result of the paper is a characterization of Prüfer domains with no non-zero idempotent prime ideals as those integral domains \(R\) for which there is a one-to-one correspondence between \(\text{Spec} R\) and the set of isomorphism classes of indecomposable injective \(R\)-modules having the property that every indecomposable injective \(R\)-module viewed as a module over its endomorphism ring, is uniserial (theorem 2.2). Further, the author shows that a generalized Dedekind domain has the property that every homomorphic image of the field of fractions of \(R\) is an injective \(R\)-module, if and only if all the localizations \(R_M\) of \(R\) at its maximal ideals are almost maximal valuation domains (theorem 4.3). Finally, by using a result of \textit{W. J. Lewis} [J. Algebra 25, 419-434 (1973; Zbl 0266.13010)] the author shows that a Lewis construction of a Bezout domain with a spectrum order isomorphic to a prescribed ordered set \(X\) is a generalized Dedekind domain, if \(X\) is a Noetherian tree with a least element (theorem 5.3).