In the context of module theory, a branch of abstract algebra, an indecomposable module is a module that cannot be expressed as a direct sum of two non-trivial submodules. More formally, a module \( M \) over a ring \( R \) is said to be indecomposable if whenever \( M \) can be written as a direct sum of two submodules \( A \) and \( B \) (i.e.
Articles by others on the same topic
A nonzero module is indecomposable if it cannot be expressed as a direct sum of two nonzero submodules. The zero module is excluded. A module of finite composition length is indecomposable exactly when its endomorphism ring has no idempotents other than zero and one: an idempotent splits the module into its image and kernel.