A uniqueness result for direct-sum decompositions: under suitable finiteness hypotheses (for example modules of finite length or modules over a semiperfect ring), any decomposition of a module into indecomposable summands is unique up to ordering and isomorphism of the summands.

Principle

Indecomposable summands behave like atomic building blocks under finiteness conditions, and direct-sum decompositions admit a cancellation/rigidity property ensuring uniqueness.

Demonstration

Let M be a finite-length module with decompositions M ≅ ⊕_{i=1}^r A_i ≅ ⊕_{j=1}^s B_j where A_i, B_j are indecomposable; Krull–Schmidt asserts r = s and, after reordering, A_i ≅ B_i for each i.

Misapplication

Assuming uniqueness of indecomposable decomposition without checking hypotheses: there exist rings and modules (without finiteness or semiperfect hypotheses) with non-unique decompositions into indecomposables.

Consequence

Enables classification of modules by their indecomposable summands, underpins structure theorems in representation theory and module categories, and allows invariants to be read off decompositions.

Reversal

In contexts lacking Krull–Schmidt hypotheses, decompositions can be non-unique and 'indecomposable' summands do not serve as canonical invariants; cancellation may fail.

Boundary

Applies under explicit finiteness/semiperfect conditions (finite length, artinian or semiperfect rings); it does not automatically hold for infinitely generated modules or arbitrary rings.

Semantic Tension

Tension with Jordan–Hölder type uniqueness (composition factors up to order) — Krull–Schmidt concerns indecomposable direct-sum factors rather than simple composition factors, and the two notions can diverge in scope and hypotheses.

Synthesis

Krull–Schmidt formalizes that, when size and ring hypotheses hold, modules decompose uniquely into indecomposable summands, making those summands the canonical atomic pieces for classification.