We have been given R, a semi-simple ring. My notes state without any justification that every R-module M is a quotient of a direct sum of copies of R. I guess this is supposed to be obvious but I can't see why this is. Thanks for any help figuring out the intermediate steps!
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1Do you mean that it is a quotient of a direct sum of copies of $R$, or that it is a direct sum of quotients of $R$? The former is true of not just modules over semi-simple rings, but modules over any ring, as your answer showed. – Joshua Mundinger May 03 '17 at 12:07
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I meant the former and I agree with you, my answer did not rely on the fact that I was dealing with a semi-simple ring R. I'll edit my question right away. – Pierre May 03 '17 at 12:18
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I think the answer to my question is as follows: (1) M is generated by itself. (2) A subset S of M generates M if and only if the homomorphism from the direct sum of R over S to M is surjective. (3) Combining (1) and (2) gives the result.
Please feel free to tell me whether this is wrong. I am new to the theory of modules and not very confident in my understanding.
Pierre
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