Editing Indecomposable module (section)

=== Principal ideal domain ===
Finitely-generated modules over [[principal ideal domain]]s (PIDs) are classified by the [[structure theorem for finitely generated modules over a principal ideal domain]]: the primary decomposition is a decomposition into indecomposable modules, so every finitely-generated module over a PID is completely decomposable.

Explicitly, the modules of the form ''R''/''p''<sup>''n''</sup> for [[prime ideal]]s ''p'' (including {{nowrap|1=''p'' = 0}}, which yields ''R'') are indecomposable. Every finitely-generated ''R''-module is a direct sum of these. Note that this is simple if and only if {{nowrap|1=''n'' = 1}} (or {{nowrap|1=''p'' = 0}}); for example, the [[cyclic group]] of order 4, '''Z'''/4, is indecomposable but not simple – it has the subgroup 2'''Z'''/4 of order 2, but this does not have a complement.

Over the [[integer]]s '''Z''', modules are [[abelian group]]s. A finitely-generated abelian group is indecomposable if and only if it is [[group isomorphism|isomorphic]] to '''Z''' or to a [[factor group]] of the form '''Z'''/''p''<sup>''n''</sup>'''Z''' for some [[prime number]] ''p'' and some positive integer ''n''. Every [[finitely-generated abelian group]] is a [[Direct sum of groups|direct sum]] of (finitely many) indecomposable abelian groups.

There are, however, other indecomposable abelian groups which are not finitely generated; examples are the [[rational number]]s '''Q''' and the [[Prüfer group|Prüfer ''p''-group]]s '''Z'''(''p''<sup>∞</sup>) for any prime number ''p''.

For a fixed positive integer ''n'', consider the ring ''R'' of ''n''-by-''n'' [[matrix (mathematics)|matrices]] with entries from the [[real number]]s (or from any other field ''K''). Then ''K''<sup>''n''</sup> is a left ''R''-module (the scalar multiplication is [[matrix multiplication]]). This is [[up to]] [[isomorphism]] the only indecomposable module over ''R''. Every left ''R''-module is a direct sum of (finitely or infinitely many) copies of this module ''K''<sup>''n''</sup>.