Basis partition gives a module direct sum ($E=E^{\prime }\bigsqcup E^{\prime \prime }⟹M=⟨E^{\prime }⟩\oplus ⟨E^{\prime \prime }⟩$)

Partition a basis of a module $E$ into disjoint subsets $E^{\prime },E^{\prime \prime }$. Splitting the finite expression of each element gives $M=⟨E^{\prime }⟩+⟨E^{\prime \prime }⟩$. An element in both generated submodules would give, by subtraction, a relation among the disjoint parts of $E$; linear independence in a module forces every coefficient to vanish. The intersection is zero, proving the direct sum of modules decomposition. Each part is itself a basis of a module for the submodule it generates.