Date of this Version
Bulletin Of The American Mathematical Society, Volume 79, Number 6, November 1973
A canonical form for a module M over a commutative ring R is a decomposition M £ R/^ © • • • 0 #//„, where the Ij are ideals oiR and I x £ ••• £ /„.A complete structure theory is developed for those rings for which every finitely generated module has a canonical form. The (possibly larger) class of rings, for which every finitely generated module is a direct sum of cyclics, is also considered, and partial results are obtained for rings with fewer than 2C prime ideals. For example, if R is countable and every finitely generated R-module is a direct sum of cyclics, then R is a principal ideal ring. Finally, some topological criteria are given for Hermite rings and elementary divisor rings.