I've leaned the $I$-adic topology and completion of a ring. I wonder if I'm understand the inverse limit correctly. Let $R$ be a commutative ring with unity and $I\subset R$ be an ideal. Then $$\varprojlim_{n}R/I^n = \{(r_i)_{i\in\Bbb N}|\forall i\leq j,r_i=f_{ij}r_j\}$$ where $(f_{ij},R/I^n)$ is an inverse system with $f_{ij}:R/I^j\to R/I^i$ is a natural projection.
One basic example of the inverse limit is that letting $R=\Bbb Z$ and $I = (p)$ for some prime number $p$. Then $$\varprojlim_{n}\Bbb Z/p^n=\bigg\{\sum_{i=0}^\infty a_ip^i\bigg|0\leq a_i\leq p-1\bigg\}$$ The form is suddenly changed to formal power series form. I think this is because of the identification $r_i =f_{ij}r_j$. I wonder if I can write this form in general:
Let $R$ be a commutative ring and $I\subset R$ be an ideal. $$\varprojlim_{n} R/I^n=\bigg\{\sum_{i=0}^\infty \overline{r}_iI^i\bigg|\overline{r}_i\in R/I\bigg\}$$ Am I write correctly? Can this be generalized in some natural way? I basically want to drop $r_i = f_{ij}r_j$ in the set. Thanks in advance.