indecomposable module
An indecomposable module is one that cannot be split into a direct sum of two smaller nonzero pieces — it comes in a single connected lump. This is a weaker, subtler notion than being simple: a simple module has no submodules at all, but an indecomposable module may be full of submodules, as long as none of them splits off as a direct summand. Indecomposables are the true building blocks for direct-sum decompositions.
A nonzero R-module M is indecomposable if M = A ⊕ B (as submodules) forces A = 0 or B = 0. Every simple module is indecomposable, but not conversely. A clean characterization: a module of finite length is indecomposable iff its endomorphism ring is local, i.e. every endomorphism is either nilpotent or an isomorphism — there are no nontrivial idempotents to project onto a summand.
The Krull-Schmidt theorem makes indecomposables decisive: a module of finite length decomposes as a direct sum of indecomposables, and that decomposition is unique up to isomorphism and reordering of the summands. Over a PID the indecomposable finitely generated modules are exactly R itself and the prime-power cyclic modules R/(p^n); the latter are indecomposable but, for n ≥ 2, far from simple, since they contain proper submodules that simply refuse to split off.
Z/4Z is indecomposable but not simple: it has the proper submodule 2Z/4Z, so it is not simple, yet it cannot be written as a direct sum of two nonzero submodules (any such decomposition would force an element of order 4 to split into elements of order ≤ 2). By contrast Z/6Z ≅ Z/2Z ⊕ Z/3Z is decomposable.
Indecomposable but not simple: Z/4Z has submodules yet no direct-summand splitting.