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Is it true that a finitely generated semisimple module $M$ a finite direct sum of simple submodules? Is it why a finitely generated semisimple module has finite uniform dimension?

I know that a semisimple module is a direct sum of simple (hence, uniform) modules. So, if the first question has positive answer, then any f. g. semisimple module would have u. dim=the number of the components in the direct summand.

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  • $\begingroup$ An infinite direct sum of nonzero modules is not finitely generated. $\endgroup$ – egreg Jul 8 '17 at 9:13
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Is it true that a finitely generated semisimple module M is a finite direct sum of simple submodules?

Of course it is. If $M$ is semisimple and f.g., the generators all lie inside a finite subset of the summands, and so their span lies there too.

Is it why a finitely generated semisimple module has finite uniform dimension?

Yes, you can conclude that.

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