indecomposable

Indecomposable is a term used in various areas of mathematics to describe objects that cannot be expressed as a nontrivial direct sum of two subobjects. In module theory, an indecomposable module M over a ring R is a nonzero module that is not isomorphic to a direct sum M1 ⊕ M2 of two nonzero submodules. The concept is common in representation theory, abelian categories, and ring theory, among others.

Indecomposability is distinct from simplicity. A simple (or irreducible) object has no proper nonzero subobjects at all, whereas an indecomposable object may contain proper subobjects but cannot be split into two nonzero summands. The endomorphism structure of indecomposables often governs their behavior in decompositions of larger objects.

Examples help clarify the idea. In abelian groups, Z/pZ is both simple and indecomposable; Z/p^2Z is indecomposable but not simple. For vector spaces over a field, every finite-dimensional space of dimension greater than 1 is decomposable into a direct sum of 1-dimensional subspaces, so the indecomposable finite-dimensional vector spaces are precisely the 1-dimensional ones. As rings, a ring R is indecomposable if it cannot be written as a direct product R1 × R2 with Ri ≠ 0; for instance, Z is indecomposable as a ring, while Z/6Z is decomposable since Z/6Z ≅ Z/2Z × Z/3Z.

A key structural result is the Krull–Schmidt theorem: in many settings (notably for modules of finite length over a suitable ring), every object decomposes uniquely as a finite direct sum of indecomposable objects, up to isomorphism and order, with indecomposable summands having local endomorphism rings.