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Our teacher told us that a finite direct sum and a finite direct product are isomorphic. Is there a simple way to prove it?

user26857
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user2435479
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    In which context? This is not true for sets, for instance. – J.-E. Pin Jan 11 '18 at 21:18
  • For a simple proof in pre-additive categories see https://math.stackexchange.com/q/2440387 – Fabio Lucchini Jan 11 '18 at 21:28
  • This is, to my mind, a more subtle issue than it appears. There is a question, for example, of in what sense the isomorphism is natural, and also in what sense it is unique. See https://qchu.wordpress.com/2012/09/14/a-meditation-on-semiadditive-categories/ for a careful discussion. – Qiaochu Yuan Jan 11 '18 at 21:36

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For $R$-modules this is true, so in particular for abelian groups. A short proof uses that the category of $R$-modules is additive. Hence finite direct sums and finite direct products coincide, because the direct sum corresponds to the coproduct, and the direct product to the product.

Reference at MSE: see the above comments, and

The direct sum $\oplus$ versus the cartesian product $\times$

Dietrich Burde
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