indecomposability

Indecomposability is a notion in algebra and representation theory describing objects that cannot be written as a nontrivial direct sum of two subobjects.

Definition: In a category with direct sums, an object A is indecomposable if every isomorphism A ≅ B ⊕ C forces either B ≅ 0 or C ≅ 0. In module theory, a module M over a ring R is indecomposable if whenever M ≅ N ⊕ P, one has N = 0 or P = 0.

Characterizations: For modules of finite length, M is indecomposable iff its endomorphism ring End_R(M) is a local ring (has a unique maximal left ideal). In general, indecomposability means there is no nontrivial internal direct sum decomposition.

Examples: A simple module over any ring is indecomposable. A one-dimensional vector space over a field is indecomposable; finite-dimensional vector spaces decompose into direct sums of 1-dimensional subspaces, so they are only indecomposable when dimension is 1.

The ring R itself, viewed as a module over itself, is indecomposable if and only if R has no nontrivial idempotents; otherwise R ≅ Re(Re) ⊕ Re′.

Theorem: Krull–Schmidt (in appropriate categories) states that objects decompose uniquely as finite direct sums of indecomposables up to isomorphism and order, provided the category satisfies the Krull–Schmidt property (e.g., finite-length modules over a ring, or finite-dimensional representations of a finite-dimensional algebra).

Applications: Indecomposability is central in the classification of representations, modules of finite length, and in the study of decomposition patterns in algebraic structures.

See also: Simple module, irreducible representation, Krull–Schmidt theorem, local endomorphism.