nondecomposable

Nondecomposable is an adjective used in mathematics and related fields to describe an object that cannot be expressed as a direct sum (or, in some contexts, a direct product) of two nonzero subobjects. In algebra, this means that whenever an object M satisfies M ≅ A ⊕ B with subobjects A and B, then either A = 0 or B = 0. The term is commonly applied to modules, representations, and other structures in which a direct-sum decomposition is a natural way to break objects into smaller pieces.

In representation theory and category theory, nondecomposable (often equated with indecomposable in many texts) contrasts with decomposable objects, which can be split into two nontrivial summands. It is important to distinguish nondecomposable from irreducible (or simple): an irreducible object has no proper nontrivial subobjects, while a nondecomposable object may have subobjects but still cannot be written as a direct sum of two nonzero subobjects.

Examples help illustrate the idea. The Z-module Z/p^2Z is indecomposable (nondecomposable) as a Z-module but not simple. A two-dimensional vector space over a field, such as k^2, is decomposable since it can be written as a direct sum of two one-dimensional subspaces. Over a local ring, certain modules, such as k[x]/(x^2) viewed as a module over itself, are indecomposable.

In Krull-Schmidt categories, objects are built from nondecomposable summands, and decompositions into such summands are unique up to isomorphism. See also indecomposable, simple, irreducible, direct sum, Krull-Schmidt theorem.