Decompositions of Modules and Matrices

Authors

Thomas Shores, University of Nebraska - LincolnFollow
Roger Wiegand, University of Nebraska - LincolnFollow

Date of this Version

11-1973

Comments

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

Abstract

A canonical form for a module M over a commutative ring R is a decomposition M ≈ R/I₁ Ο … Ο R/I_n, where the I_j are ideals of R and 1₁ ≤ . . . ≤ 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.