Decompositions Of Modules And Matrices

Authors

Thomas S. Shores, University of Nebraska-Lincoln
Roger Wiegand, University of Nebraska-LincolnFollow

Document Type

Article

Date of this Version

1973

Citation

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

Comments

Copyright American Mathematical Society 1974

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.