Indecomposable module

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.