I have a question concerning example 3.14 of Hatcher's Algebraic Topology. There are a few details that confuse me, so I suppose the goal of my questions is more or less clarification - I apologize if they seem trivial. The first (ring)-isomorphism written is $$H^*(\coprod_\alpha X_\alpha;R)\stackrel{\cong}{\longrightarrow}\prod_\alpha H^*(X_\alpha;R)\quad(*)$$ Have I understood it correctly when I say that the argument is based on the fact that the inclusions $i:X_{\alpha_0}\hookrightarrow\coprod_\alpha X_\alpha$ induce, on the level of groups, the isomorphism $(*)$ and since continuous maps induce morphisms that respect the $\cup$-product, then $(*)$ is in addition in fact a ring-isomorphism?
If so, then I can not quite understand why we need he additional assumption on the pair $(X_\alpha,x_\alpha)$ when arguing that we have an isomorphism similar to $(*)$ for the wedge sum. Consider the following reasoning: On the level of groups we have $$H^*(\bigvee_\alpha X_\alpha;R):= \bigoplus_{i=0}^\infty H^i(\bigvee_\alpha X_\alpha;R)\stackrel{\prod i^*}{\to}\prod_\alpha\bigoplus_{i=0}^\infty H^i(X_\alpha;R)=\prod_\alpha H^*(X_\alpha;R)$$ where $\prod i^*$ is induced by the inclusion forming an isomorphism by the Eilenberg-Steenrod axiom for the wedge sum. Again, since continuous maps induce ring-morphisms, the isomorphism is in fact a ring-isomorphism and the claim follows. Of course something is wrong because I do not apply the additional assumption Hatcher makes. If anyone would care to point out where my mistake is and explain how one implement the assumption correctly, I would appreciate that very much. Thank you in advance!
