 The primary decomposition generalizes to finitely generated modules over commutative Noetherian rings, and this result is called the Lasker–Noether theorem.By contrast, unique decomposition into "indecomposable" submodules does not generalize as far, and the failure is measured by the ideal class group, which vanishes for PIDs.For rings that are not principal ideal domains, unique decomposition need not even hold for modules over a ring generated by two elements.