Dimirreducible

Dimirreducible is a term used in dimension theory and category theory to describe objects that cannot be decomposed into nontrivial direct sums without changing their dimension. In a category equipped with a dimension function that is additive on direct sums, an object X is called dimirreducible if X is nonzero and whenever X is isomorphic to a direct sum A ⊕ B, either A is zero or B is zero. Equivalently, X has no nontrivial direct sum decomposition compatible with the given dimension function.

Relation to indecomposability: In many standard settings, dimirreducible coincides with the usual notion of indecomposable (also called irreducible) objects, which are those not isomorphic to a direct sum of two nonzero objects. The dimension function provides an additional organizing principle, but does not alter the basic indecomposability criterion in additive categories.

Examples: In the category of finite-dimensional vector spaces over a field with the ordinary dimension, the dimirreducible objects are exactly the one-dimensional spaces. A 1-dimensional space cannot be nontrivially split as a direct sum of nonzero subspaces. By contrast, any vector space of dimension two or higher can be written as a direct sum of subspaces, so it is not dimirreducible.

Variants and generalizations: The concept extends to any category with an additive dimension function valued in the natural numbers or a commutative monoid, and to more general decompositions that respect the dimension structure. Some authors distinguish dimirreducible from strictly indecomposable when additional constraints are imposed on subobjects or morphisms.

See also: indecomposable, simple object, Krull-Schmidt theorem, dimension theory.