On Aug 24, 5:33 am, "Nicolas M. Thiery" <nicolas.thi...@u-psud.fr>
wrote:
> - The problem with the map a -> (a,0) is only that 1_A is mapped to
> (1_A,0) which is not 1_{A\oplusB} = (1_A,1_B), right?
>
> Otherwise said, the category of NonUnitalAlgebras (which is not yet
> implemented in Sage) indeed has a direct sum?
I don't think so, but I could be wrong. There are maps of nonunital
algebras from each of A and B to A \oplus B, but I don't think A
\oplus B has the correct universal property: given ring maps f:A --> C
and g: B --> C, you want a map
h: A \oplus B --> C
compatible with f, g, and the inclusions of A, B into A \oplus B.
Since (a,b) = (a,0) + (0,b), I think the only way to define h is as h
(a,b) = f(a) + g(b). But this isn't compatible with the product
structure: in general,
h((a1,b1) (a2, b2)) != h(a1,b1) h(a2,b2).
Either that, or I'm being completely silly (or cocompletely silly,
since we're talking about coproducts).
Regards,
John
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