Page "Indecomposable module" Paragraph 13

Explicitly, the modules of the form for prime ideals p ( including 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 n = 1 ( or p = 0 ); for example, the cyclic group of order 4, Z / 4, is indecomposable but not simple – it has the subgroup 2Z / 4 of order 2, but this does not have a complement.