Two submodules form an internal direct sum exactly when they span the module and intersect trivially.Suggest a slogan
Let \(R\) be a ring, \(M\) an \(R\)-module, and \(M_1, M_2 \leq M\) submodules. The \(R\)-linear map \(M_1\oplus M_2\to M\), \((x,y)\mapsto x+y\), is an isomorphism, so that \(M = M_1 \oplus M_2\) internally, if and only if
\(M_1 + M_2 = M\), and
\(M_1 \cap M_2 = 0\).
The image of the map is \(M_1+M_2\), so it is surjective if and only if \(M_1+M_2=M\). Its kernel is \(\theset{(x,-x)\suchthat x\in M_1\cap M_2}\), so it is injective if and only if \(M_1\cap M_2=0\).