A question on decomposition of modules

Question

Let $M$ be an $R$-module. A decomposition $M=⊕_A M_\alpha$ of nonzero submodules is said to complement direct summands in case for each direct summand $K$ of $M$ there is a subset $B$ of $A$ with $M=(⊕_B M_\beta)⊕K$. Is it true that each $M_\alpha$ is necessarily indecomposable?

Answer 1

Isn't this obvious?

If $K$ is a direct summand of $M_{\alpha}$, then $K$ is a direct summand of $M$, so $M=(⊕_B M_\beta)⊕K$. We must have $\alpha\notin B$, so $B\subsetneq A$. If $x\in M_{\alpha}$, then $x\in M$, and therefore $x=y+z$ with $y\in ⊕_B M_\beta$ and $z\in K$. But then $y\in M_{\alpha}$, so $y=0$.