In mathematics, in the field of abstract algebra, the structure theorem for finitely generated modules over a principal ideal domain is a generalization of the fundamental theorem of finitely generated abelian groups and roughly states that finitely generated modules can be uniquely decomposed in much the same way that integers have a prime factorization. The result provides a simple framework to understand various canonical form results for square matrices over fields.
... Similarly for modules that are not finitely generated, one cannot expect such a nice decomposition even the number of factors may vary ... There are Z-submodules of Q4 which are simultaneously direct sums of two indecomposable modules and direct sums of three indecomposable modules, showing the analogue of the primary ... Another issue that arises with non-finitely generated modules is that there are torsion-free modules which are not free ...
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