In chapter 4 of Milnor/Stasheff's Characteristic Classes, the authors define the ring $H^\prod(B; \mathbb{Z}/2)$ to be the ring of formal infinite series: $a = a_0 + a_1 + a_2 + \cdots$ , where $a_i \in H^i(B; \mathbb{Z}/2)$. Addition and multiplication in this ring are defined as one would expect: by adding or multiplying the formal series.
Just to check my understanding of this construction, I have two questions:
If the cohomology of $B$ vanishes above some dimension, say, $H^i(B; \mathbb{Z}/2) = 0$ for $i > n$, then is $H^\prod(B; \mathbb{Z}/2)$ simply isomorphic to the usual cohomology ring $H^*(B; \mathbb{Z}/2)$?
In general, if $R$ is an $\mathbb{N}$-graded ring, it seems that the same construction be carried out, creating a ring $R^\prod$ of formal infinite series. If so, is the assignment $R \mapsto R^\prod$ functorial (from the category of $\mathbb{N}$-graded rings to the ordinary category of rings)? And is $R^\prod$ the usual notation for this?