## Mathematics, Department of

#### Date of this Version

11-1973

#### Abstract

A canonical form for a module *M* over a commutative ring *R* is a decomposition *M* ≈ *R*/*I*_{1} Ο … Ο *R*/*I*_{n}, where the *I*_{j} are ideals of *R* and 1_{1} ≤ . . . ≤ *I _{n}*. 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

Published in

BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETYVolume 79, Number 6, November 1973. Copyright © American Mathematical Society 1974. Used by permission.