Our teacher told us that a finite direct sum and a finite direct product are isomorphic. Is there a simple way to prove it?
2026-03-30 20:44:24.1774903464
How to show that a finite direct sum is isomorphic to a finite direct product?
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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 $\times$