I'm looking at a statement in Rotman's 'Introduction to Homological Algebra' which I'm having a problem with:
Theorem 2.30.i: There is a $Z(R)$-isomorphism $$\varphi : \text_R \bigg( A, \prod_ B_i \bigg) \to \prod_ \text_R (A, B_i)$$ with $\varphi : f \mapsto (p_if)$, where the $p_i$ are the projection of the direct product $\displaystyle \prod_ B_i$. If $R$ is commutative, then $\varphi$ is an $R$-isomorphism.
Q1: I don't understand why we need to restrict it to a $Z(R)$-map? Q2: It' simply an $R$-map, isn't it? which makes the last line in the statment redundant? In his proof, he says the following:
To see that $\varphi$ is a $Z(R)$-map, note, for each $i$ and each $r \in Z(R)$, that $p_i r f = r p_i f$; therefore, $$\varphi : rf \mapsto (p_i r f) = (rp_i f) = r(p_if) = r \varphi(f).$$
But I don't see how he's using anything other than the fact that $p_i$ is an $R$-map which allows us to pull out the scalar. Q3: Am I missing something here? Or did he add an unnecessary hypothesis?