Suppose, we do universal algebra in a "non-evil" fashion, such that every algebra carries around an equivalence relation $\cong$ that poses as equality and such that size issues don't matter (we may have proper classes as underlying "sets").
If we consider the category of finite sets with binary products $\times$, binary coproducts $+$, the empty set $0 := \emptyset$ and some one-element set $1$ together with isomorphy (is that a word?) of objects $\cong$, we get the rig (or semiring if you will) of natural numbers.
Are there "nice" and "interesting" categories equipped with finite products and coproducts that in the same fashion yield the ring of integers or the field of rationals?
I'm afraid, that this question may have a really easy answer, however...