(nondirected) Colimit in the category of rings

Dear NG,

This is something that bugs me every now and then. I haven't tried
giving it a serious thought. But I think there must be known reference
or construction on colimits in the category of rings that is not from a
directed diagram. I know exactly how I can construct directed colimit,
I have seen them in books.. but with arbitrary colimits I get confused.
In books the typical way of constructing colimit is to take the direct
sum of the group obtained from the ring (in our diagram) with addition
and then take a residue modulo some subgroup (here is when the
directedness of the diagram is used.. the residue namely takes away an
element difference with the directed diagram mappings of it) and then
define a multiplication of this group to make a ring which will turn
out to be a directed colimit.

As for the nondirected colimits, the category theory books I have just
say that the category of rings is cocomplete (meaning that there is a
colimit), but an exact construction is not available. Probably using
tensor product is a natural way of constructing it?

Sincerely,
Jose Capco

.

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