Is the sum of a family of simple submodules always interal direct sum?

Question

Let $\lbrace S_i \rbrace_{i\in I}$ be a family of pairwisw disjoint non-isomorphic simple submodules of a module $M_R$. Is it true that the sum $\sum_{i\in I}S_i$ is internal direct sum (that is, $S_k \cap \sum_{i\neq k}S_k =0$). If so, how could I prove it?. If not, my textbook writes $\bigoplus_{i\in I} S_i \subseteq M$ but this is wrong since the sum here refers to the external direct sum which is a subset of $M^I$. Am I mistaken? Can anyone clarify this for me?

I need any help. Thanks in advance.

Well, $S_k\cap A$ is a submodule of $S_k$ for any submodule $A$. — Berci, Nov 03 '21 at 07:51
We only need to exclude the case $S_k\subseteq \sum_{i\ne k}S_i$. If $S_i$'s are assumed pairwise nonisomorphic then it follows by considering the induced homomorphisms $S_k\to S_i$. — Berci, Nov 03 '21 at 08:10
Actually, $S_i$'s are assumed pairwise nonisomorphic. Can you please clarify what you mean? @Berci — Hussein Eid, Nov 03 '21 at 08:13
Kindly, clarify your answer. Up till now, I didn't get what you meant. Thanks — Hussein Eid, Nov 03 '21 at 08:42

score 1 · Accepted Answer · answered Nov 03 '21 at 09:29

Let $A_k=\sum_{i\ne k}S_i$.

By simplicity of $S_k$ we only need to exclude the case $S_k\subseteq A_k$ because $S_k\cap A_k\in\{0, S_k\}$.

First we can prove the claim for finite $I$ by induction, so the induction hypothesis implies $A_k=\bigoplus_{i\ne k}S_i$ and thus we'd obtain a homomorphism $S_k\hookrightarrow A_k\to S_i$ for each $i\ne k$ which together determine $S_k\hookrightarrow A_k$, but these all must be trivial.

For infinite $I$, assume $x\in S_k\cap A_k$. Then, $x=s_{i_1}+s_{i_2}+\dots+s_{i_r}$ for some finite index set $\{i_1,\dots,i_r\}\subseteq I\setminus\{k\}$ therefore $x\in S_k\cap\sum_{u\le r} S_{i_u}$ leading back to the finite case, implying $x=0$.

Is the sum of a family of simple submodules always interal direct sum?

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