Mathematics, Department of
Department of Mathematics: Faculty Publications
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Document Type
Article
Date of this Version
11-1973
Abstract
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.
Comments
Published in BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY Volume 79, Number 6, November 1973. Copyright © American Mathematical Society 1974. Used by permission.