I've just seen on Wikipedia that we can't speak of direct sum of rings. Let $R$ and $S$ be rings. It says we can't have a direct sum of rings because the direct sum $R\times S$ doesn't receive a natural ring homomorphism. I don't understand what it means by this, and why we need this natural ring homomorphism to have direct sum of rings.
2026-03-27 04:23:34.1774585414
Why isn't there necessarily a direct sum of rings?
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There is no natural ring homomorphism that maps $1$ to $(1,1)$ in the direct sum.
However projections onto the factors are natural ring homomorphisms, as they map $(1,1)$ onto $1$.