Hi, I keep seeing indirect uses of a result which I think would be stated as follows:(adsbygoogle = window.adsbygoogle || []).push({});

If a module [itex]M[/itex] over the unital associative algebra [itex]A[/itex] is written

[itex]M\cong S_1\oplus\cdots\oplus S_r[/itex] (where the [itex]S_i[/itex] are simple modules), then in any comosition series of [itex]M[/itex], the composition factors are, up to order and isomorphism, [itex]S_1,\ldots,S_r[/itex]. Perhaps this statement is not correct, but it's the best I can do when I can't find an explicit statement anywhere.

I think I would use the Jordan Holder theorem to prove this. I would argue as follows: a composition series for [itex]S_1\oplus\cdots\oplus S_r[/itex] is given by

[itex]S_1\oplus\cdots\oplus S_r>S_1\oplus\cdots\oplus S_{r-1}>S_1\oplus\cdots\oplus S_{r-2}>\cdots>S_1>\{0\}.[/itex]

Now, [itex](S_1/\{0\})\cong S_1,\ (S_1\oplus S_2)/S_1\cong S_2,\ (S_1\oplus S_2\oplus S_3)/(S_1\oplus S_2)\cong S_3,\ldots, (S_1\oplus\cdots\oplus S_r)/(S_1\oplus\cdots\oplus S_{r-1})\cong S_r[/itex] and so by the Jordan Holder theorem, in any comosition series of [itex]M[/itex], the composition factors are, up to order and isomorphism, [itex]S_1,\ldots,S_r[/itex].

I have a feeling that I am going wrong somewhere, perhaps in my cancelling when I do things like [itex] (S_1\oplus S_2)/S_1\cong S_2[/itex]. The terminology of "direct sum" would suggest that this is not allowed.

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# Quotients of direct sums of modules

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