The infinite Direct Sum in the category Ring

If you don't have strong personal feelings about it already, most of you have at least witnessed the opposing factions on how we should define a ring and, by extension, how we should define a ring homomorphism. The strong majority define a ring to have a unity and require that ring homomorphisms preserve it. Others do not require these things of a ring.

The general pattern I've seen which determines into which camp any given mathematician falls is how 'good' they want their category of rings to be. Those who want zero morphisms and want to be able to define the kernel of a morphism categorically often opt to go without unity. There is a printed semi-famous comment about this problem in the Radical Theory of Rings by Gardner and Wiegandt:

This is where my question begins. I have searched for why the category of Rings (with unity) does not have infinite coproducts. And basically everywhere that comments on it does not prove that there is no object in Ring which satisfies the universal property of a coproduct for an infinite family.

What they do explain is that the thing we would expect to be the coproduct (namely the set-theoretic direct sum with pointwise addition and multiplication) is not a coproduct in Ring.

I have attempted to prove the non-existence of such an object for myself but have not gotten far. One would have to show that either every infinite family of rings cannot all be embedded in another ring; or in the case that there is such an object, there is always a 'smaller' (non-isomorphic) object in Ring which has the same property.

Thus my question is can someone prove to me that Ring (the category of rings with unity and unity preserving homomorphisms) does not have infinite coproducts? Or can you direct me to a reference that may have my answer?

It does not say there is nothing satisfying the coproduct axioms: As you say, it is just saying that the direct sum construction (the elements of the Cartesian product which are finitely nonzero) is a ring without multiplicative identity. This is straightforward to show.

In fact ring categories can and do have objects satisfying the coproduct axioms. It's just that they aren't the things given by the direct sum construction. The right thing is built with a sort of free product of rings.

In fact you will probably be satisfied by Qiaochu's answer here which goes into a little detail about it for someone asking for an explicit construction. Martin Brandenburg has also supplied a detailed construction in a comment below.