On a question concerning $D4$-modules

An $R$-module $M$ is called a $D4$-module if ‘whenever $M_1$ and $M_2$ are direct summands of $M$ with $M_1 +M_2 = M$ and $M_1 \cong M_2$, then $M_1 \setminus M_2$ is a direct summand of $M'$. Let $M = \oplus_{i \in I} M_i$ be a direct sum of submodules $M_i$ with $H_om(M_i, M_j) = 0$ for distinct $i$, $j \in I$. We show that $M$ is a $D4$-module if and only if for each $i \in I$ the module $M_i$ is a $D4$-module. This settles an open question concerning direct sums of $D4$-modules. Our approach is independent of the solution obtained by D’Este, Keskin Tütüncü and Tribak recently.