Dedekind-like behavior of rings with 2-generate ideals

Abstract

A Bass ring is a commutative ring without nilpotent elements and with module fini~:e integral closure, such that every ideal is generated by two elements. The~ rings have received attention from several points of view algebraic number theor( representation theory and algebraic geometry and have been called Bass orde when they are orders in separable algebras over the quotient field of a Dedeki~ domain [6, §37]. See Section 2 for more comments about where Bass rings aris. Bass, in his 1963 paper [2], showed that, as in the case of Dedekind domain every finitely generated torsionfree module over a Bass ring R is isomorphic to direct sum of ideals of R. Our main result is a uniqueness theorem that furth, extends the analogy with Dedekind domains. We use the representation-theoret term 'R-lattice' to mean 'finitely generated torsionfree R-module ' . An R-lattic then, is a finitely generated R-submodule of a K-module, where K is the classic quotient ring of R. With each R-lattice we associate an isomorphism class d(M) (the ideal class of 21, oi faithful ideals of R, and then prove: