Distributivity of direct sum over Hom Let's suppose we are…
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Distributivity of direct sum over Hom
Let's suppose we are working in the category of left R-modules (R can be noncommutative)? If I am correct, why do we have Hom(A, B \\oplus C) = Hom(A,B) \\oplus Hom(A,C), but not Hom(A \\oplus B,C) = Hom(A,C) \\oplus Hom(B,C)?
What is an example showing why the second property is not true?
I think direct sum in the first variable sends direct sums to products, while direct sum in the second variable preserves direct sums. I know this has something to do with abelian categories more generally and (co)products, but I have no intuitive understanding of co(products). Also, I don't really get why the difference between direct sum and product is so significant, given direct sum is just the product but only finitely many components can be nonzero.
#science
Let's suppose we are working in the category of left R-modules (R can be noncommutative)? If I am correct, why do we have Hom(A, B \\oplus C) = Hom(A,B) \\oplus Hom(A,C), but not Hom(A \\oplus B,C) = Hom(A,C) \\oplus Hom(B,C)?
What is an example showing why the second property is not true?
I think direct sum in the first variable sends direct sums to products, while direct sum in the second variable preserves direct sums. I know this has something to do with abelian categories more generally and (co)products, but I have no intuitive understanding of co(products). Also, I don't really get why the difference between direct sum and product is so significant, given direct sum is just the product but only finitely many components can be nonzero.
#science
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