Krull–Schmidt theorem for modules

Statement

If $E \neq 0$ is a module that satisfies the ACC and DCC on submodules (that is, it is both Noetherian and Artinian or – equivalently – of finite length), then $E$ is a direct sum of indecomposable modules. Up to a permutation, the indecomposable components in such a direct sum are uniquely determined up to isomorphism.

Goro Azumaya generalised this, showing that if a module has a (possibly infinite) decomposition into strongly indecomposable modules, then this decomposition is unique up to a permutation and isomorphism. A module is strongly indecomposable if its ring of endomorphisms is local. Any such module is indecomposable, but the converse is not true, as witnessed by the ring of integers viewed as a module over itself. Indecomposable injectives and indecomposable modules of finite length are strongly indecomposable, as are indecomposable projective modules over semiperfect rings.

In general, the theorem fails if one only assumes that the module is Noetherian or Artinian.