Indecomposable module

An indecomposable module is a term from the mathematical field of algebra. There is a special class of modules, which can not be decomposed into a direct sum. Under certain conditions, one can show that every module is a direct sum of indecomposable modules (see Theorem of Krull - Remak - Schmidt). However, there are rings, and the modules, for which this is not the case.

Definition

A - module over a ring is called irreducible, if not as a direct sum of two leaves of non-zero -modules and write.

This definition carries over mutatis mutandis to arbitrary abelian categories.

Examples

• A vector space over a field if and only indecomposable if it is one-dimensional.
• Every simple module is indecomposable, but not vice versa.
• A module of finite length if and only indecomposable if its endomorphism ring is local.