Re: connected categories and epimorphisms.

[Nikita Danilov <Nikita@Namesys.COM>]

Peter Freyd writes:
 > The quickest natural example I know of a connected category in which
 > projections from products needn't be epi is the category of
 > commutative rings. Well, actually, the opposite category.  The
 > coproduct of  Z_2  and  Z_3  is the terminal ring. The two
 > co-projections fail to be monic (the coproduct of a pair of objects in
 > this category is their tensor product).
 >

It is interesting that construction of the coproduct in the category of
"just" rings (associative, unitary, but not necessary commutative) is
not easy to find in the literature.

It seems, however, that it is relatively easy to construct: for given
rings R1 and R2, let M1 and M2 be their multiplicative monoids and M be
coproduct of M1 and M2 in the category of monoids. Form monoid ring of M
over Z. That is, form free abelian group generated by underlying set of
M and extend multiplication from generators by distributivity. Let's
call resulting ring G. Let K be ideal in G generated by all elements of
the form

  (A + B) - A - B,

where both A and B belong to the same ring: either R1 or R2. That is, in
((A + B) - A - B) "+" is addition in R1 or R2, and "-" is subtraction in
G.

Now, quotient G/K is a coproduct of R1 and R2 in the category of rings, which
is easy to prove using universal properties of coproduct M, monoid ring G, and
epicness of the projection G ->> G/K.

Is there more "direct" description of coproduct in the category of rings?

 >
 >

Nikita.