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 2 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. All rings in this announcement are commutative with 1, and all modules are unital. A canonical form for an K-module M is a decomposition M ^ Rjlx © • • • © R/In, where Ix g • • • g In # jR.IfM has a canonical form, the ideals Ij are uniquely determined [K]. A CF-ring is a ring for which every finitely generated direct sum of cyclics has a canonical form. It can be shown that R is CF if and only if R/I © R/J S R/(I nJ)® R/(I + J) for every pair of ideals /, J. By a valuation ring we shall mean a ring, possibly with zero-divisors, whose lattice of ideals is totally ordered. A ring R is arithmetical, provided the local ring Rm is a valuation ring for each maximal ideal m. Finally, an Zi-local domain [ M l ] is an integral domain such that (1) every nonzero ideal is contained in only finitely many maximal ideals, and (2) every nonzero prime ideal is contained in a unique maximal ideal. THEOREM 1. Every CF-ring is a finite direct product of indecomposable C F-rings. The indecomposable CF-ring s are precisely the rings R such that (i) R is arithmetical, (ii) R has a unique minimal prime P, (iii) R/P is an h-local domain, and (iv) every ideal contained in P is comparable with every ideal of R. Thus valuation rings and arithmetical /z-local domains are CF-rings. AMS (MOS) subject classifications (1970). Primary 13C05, 13F05; Secondary 15A21.