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A canonical form for a module M over a commutative ring R is a decomposition M ≈ R/I1 Ο … Ο R/In, where the Ij are ideals of R and 11 ≤ . . . ≤ In. 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.