What is a $R$ adic completion of $R$?

Question

Let $R$ be a ring. $R$ is an ideal of $R$, then, what is the completion of $R$ with respect to ideal $R$?

I know completion by trivial ideal $0$ is a ring itself. Is the result 0? But we know $A$⊂completion of $A$. Thank you for your help.

Answer 1

If $I$ is an ideal of $R$, the completion $\hat{R}$ of $R$ at $I$ is the inverse limit of the system

$$R/I \leftarrow R/I^2 \leftarrow R/I^3 \leftarrow \cdots .$$

Yes, if $I = R$, then each of the rings $R/I^n$ is the zero ring, and $\hat{R} = 0$.

No, a ring is not necessarily a subring of its completion. There is a canonical ring homomorphism $R\rightarrow \hat{R}$, but this homomorphism is injective if and only if $\bigcap\limits_{n=1}^{\infty}I^n = 0$.