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February 5th, 2018, 03:20 PM   #1
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Family of modules

Please help me to prove the following result:

Let $R$ be a ring with $1$ and $\mathcal{F}$ a family of simple left $R$ modules.

Let $M=\oplus_{S\in \mathcal{F}} S$ and suppose that $T$ is a simple submodule of $M$.

Show that $T\cong S$ for some $S\in \mathcal{F}$.

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February 6th, 2018, 12:47 AM   #2
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Maybe do an easy case of $M=S_1\oplus S_2$ first?

Take $T$ a submodule of $M$, you can distinguish some cases:
1) $T$ contains only elements from $S_1$.
2) $T$ contains only elements from $S_2$.
3) $T$ contains nonzero elements from both $S_1$ and $S_2$.

Now continue.
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February 6th, 2018, 01:20 AM   #3
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Micrm@ss, In this case we need to show that $T\cong S_1$ or $T\cong S_2$ but i don't see how to do so. Would you please help me more? thanks in advance
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February 7th, 2018, 08:49 AM   #4
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Mona,
Since you never answer my hints or proof outlines, here's a complete proof:

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February 7th, 2018, 11:03 AM   #5
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at the end of your proof the sum + are they simple sum or direct sum $\oplus$? and why $T\cong (M_{n-1}\oplus T)/M_{n-1}$?

Last edited by mona123; February 7th, 2018 at 11:14 AM.
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February 9th, 2018, 05:28 PM   #6
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The sum is direct, but one doesn't need this. For any submodules A and B,
$${A+B\over A}\simeq {B\over A\cap B}$$
In particular, for $A\cap B=0$, (here $0$ means the zero submodule),
$${A+B\over A}\simeq {B\over 0}\simeq B$$
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