I don't understand this problem:
Let $M_1,M_2,\dots,M_r\leq M$ submodules of the module $M$. Then $$M=M_1\oplus M_2\oplus \dots \oplus M_r$$ if and only if $M=M_1+M_2+\dots M_r$ and for each $i$, we have that $$M_i\cap (M_1+ M_2+\dots + \hat{M_i} \dots + M_r) = 0$$
If I have that $M=M_1\oplus M_2\oplus \dots \oplus M_r$ is not true by construction that $M_i\cap (M_1+ M_2+\dots + \hat{M_i} \dots + M_r) = 0$?
If I have all those intersections and the fact that $M=M_1+M_2+\dots M_r$, I can't make an isomorphism between $M$ and the product of the $M_i$?
I'm making something wrong or I'm not considering something? How would you do this problem?
Everything depends on the basic definitions you work with so let me present the definitions under which the question makes sense.
Definition 1: Given a module $M$ and a collection $\{M_i\}_{i \in I}$ of submodules of $M$, we say that $M$ is the internal sum of the $M_i$ and write $M = \sum_{i \in I} M_i$ (or $M = M_{i_1} + \dots + M_{i_N}$ if $I = \{ i_1, \dots, i_N \}$ is finite) if every element $m \in M$ can be written as a sum $m = \sum_{i \in I} m_i$ for $m_i \in M_i$ where only finite many $m_i$'s are non-zero. We say that $M$ is the internal direct sum of the $M_i$ and write $M = \oplus_{i \in I} M_i$ if every element $m \in M$ can be written uniquely as a sum $m = \sum_{i \in I} m_i$ for $m_i \in M_i$ where only finite many $m_i$'s are non-zero.
Definition 2: Given a family of modules $\{ M_i \}_{i \in I}$ which are now not assumed to be submodules of a common module, we can construct a module $M := \oplus_{i \in I} M_i$ which is called the (external) direct sum of the modules $M_i$ by setting $$ M = \{ (m_i)_{i \in I} \, | \, m_i \in M_i \text{ for all } i \in I, m_i \neq 0 \text{ only for finitely many } i \in I \} $$ and defining the obvious addition and multiplication operations. Each $M_i$ can be embedded inside $M$ using the map $\psi_i \colon M_i \rightarrow M$ which sends $m \in M_i$ to a sequence whose $i$-th index is $m$ and whose $j$-th index for $j \neq i$ is $0$.
The unfortunate thing is that the internal direct sum and the external direct sum are denoted by the same symbol. This is not so bad as they are naturally isomorphic with the external direct sum $\bigoplus_{i \in I} M_i$ being the internal direct sum of $\psi_i(M_i)$ but this leads to some confusion.
Using the definitions above, you are then asked to show that $M$ is the internal direct sum of the $M_i$ if and only if $M$ is an internal sum and you have the intersection property.