Modules for quadratic extensions of Dedekind rings

D. K. Faddeev

Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 8, Tome 160 (1987)

Résumé

Let $\sigma$ be a Dedekind ring, let $Q$ be a maximal order in a quadratic extension $K$ of the field $k$ of quotients of the ring $\sigma$, let $\Lambda$ be a subring of the ring $\sigma$, containing $\sigma$ and such that $\Lambda k=K$. It is proved that $\sigma/\Lambda$is a cyclic $\Lambda$-module. From here there follows, in particular, that each finitely generated torsion-free $\Lambda$-module is a direct sum of modules which are isomorphic to the ideals of ring $\Lambda$.