Definition 3

— A left module M over a ring A is said to be decomposable if it is the direct sum of a family of proper nonzero submodules. Otherwise it is said to be indecomposable. The zero module is thus decomposable, being the direct sum of the empty family of submodules. Let $\mathfrak{a}$ be a left ideal of the ring A; the submodules of $A/\mathfrak{a}$ are just the quotients $b/\mathfrak{a}$, where $b$ is an ideal of A containing $\mathfrak{a}$ (I, p. 41, Th. 4); if $b$ and $c$ are two ideals of A containing $\mathfrak{a}$, then the module $A/\mathfrak{a}$ is the direct sum of the submodules $b/\mathfrak{a}$ and $c/\mathfrak{a}$ if and only if $A = b + c$ and $b \cap c = \mathfrak{a}$. As a result: