Commutative orders whose lattices are direct sums of ideals (Q1107581)

Let R be a commutative semiprime Noetherian ring with Krull dimension \(\leq 1\) and with module-finite integral closure. An R-lattice is a finitely generated submodule of a free R-module. The ring R is called a \(\Sigma\) I ring provided every R-lattice is isomorphic to a direct sum of ideals of R. The aim of this paper is to determine when R is a \(\Sigma\) I ring. If every ideal of R is 2-generated then R is a \(\Sigma\) I ring [\textit{H. Bass}, Math. Z. 82, 8-28 (1963; Zbl 0112.266)].\ \textit{L. A. Nazarova} and \textit{A. V. Rojter}, Soviet Math., Dokl. 8(1967), 1089-1092 (1968); translation from Dokl. Akad. Nauk SSSR 176, 266-268 (1967; Zbl 0176.317)] determined all local \(\Sigma\) I rings. \textit{C. Greither} [J. Pure Appl. Algebra 24, 265-276 (1982; Zbl 0522.13009)] showed that there exist locally \(\Sigma\) I rings (i.e. every localization at a maximal ideal is a \(\Sigma\) I ring) which are not \(\Sigma\) I. The authors determine which locally \(\Sigma\) I rings are \(\Sigma\) I rings in terms of a graph-theoretic condition on spec(R). Secondly they determine which local rings R are \(\Sigma\) I rings in terms of certain pullbacks. The authors give an example of a locally \(\Sigma\) I ring with a ``very large'' indecomposable lattice and also geometric examples of local \(\Sigma\) I rings. Finally it is proved that if R is a \(\Sigma\) I ring then every ideal of R can be generated by three elements.