Fix a ring \(R\) and call the left \(R\)-modules simply modules. Call (as usual) a module \(M\) distributive if \(S \cap (T + U) = S \cap T + S \cap U\) for all submodules \(S\), \(T\) and \(U\) of \(M\). The aim of the paper is to prove the following two theorems. Theorem A: A direct sum of modules \(\oplus_{i\in I} M_i\) is distributive iff \((1)\) every term \(M_i\) is distributive and \((2)\) every submodule \(S\) of \(\oplus_{i\in I} M_i\) splits with respect to \(\bigoplus_{i\in I} M_i\) (that is, \(S=\oplus_{i\in I} S_i\) for some submodules \(S_i\) of \(M_i\)). As usual, a simple subquotient of a module \(M\) is a simple module of the form \(S/T\) where \(T\subseteq S\) are submodules of \(M\). The authors call two modules \(M\) and \(N\) orthogonal if no simple subquotient of \(M\) is isomorphic to a simple subquotient of \(N\). Theorem B states that condition \((2)\) of Theorem A holds iff \(M_i\) is orthogonal to \(M_j\) for all distinct \(i,j\in I\).