## Mathematics, Department of

#### Date of this Version

1973

#### Citation

Bulletin Of The American Mathematical Society, Volume 79, Number 6, November 1973

#### Abstract

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^{C} 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.

## Comments

Copyright American Mathematical Society 1974