Let $(\prod A_i,\gamma_{jk}$) be an inverse system of rings and ring homomorphisms.
Show that if all $A_i$ are commutative rings with 1 and all $\gamma_{ij}$ are ring homomorphisms that send 1 to 1, then P may likewise be given the structure of a commutative ring with 1 such that all $\gamma_i$ are ring homomorphisms.
$\gamma_i:P \rightarrow A_i$ is the projection map where P is the subring of $\prod A_i$ s.t. $\gamma_{ji}(a_j)=a_i$ whenever $j \leq i$.
Sidequestion: As long as I wrote all this up, I guess I might as well ask another question that I feel like I already proved (rather informally) and was wondering if anyone could tell me how to do it properly, I was thinking some sort of induction. But my other question is how exactly do we show that $P$ is a sub ring of the direct product? and really if I could even just get help showing it was an (abelian) group then I would be satisfied. I was thinking something like this:
I want to show that if $a_i,b_i \in A_i$ then $a_i-b_i \in A_i$. So let $k \leq i$ and then we want to show that
$\gamma_{ik}(a_i-b_i) \in A_k$, and we get to assume that $\gamma_{ik}$ are group homomorphisms.
Anyway, then $\gamma_{ik}(a_i-b_i) = \gamma_{ik}(a_i) - \gamma_{ik}(b_i)$.
So now we have to show $\gamma_{ik}(a_i) - \gamma_{ik}(b_i) \in A_k$... repeat the process? I don't know anything about the indexing set so I can't get to the end of the process... so yeah what do I do here? Thanks in advance!