I've to prove the following theorem:

> 
> Let R a ring, I one of its two-sided ideal and π: R ⟶ R/I, the
> canonical projection. Then
> 
> 
> * S subring (ideal) of R ⇒ π(S) = (S + I)/I is a subring
> (ideal) of R/I;
> * S subring (ideal) of R/I ⇒
> π−1(S') is a subring (ideal) of R containing I.
> 
> 
> 

The map $\pi$ is $r ⟼ r + I$ with $r \in R$, so considering $s ⟼ s + I$ with $s \in S$, it should follow S + I ⊂ R + I, since S ⊂ R.

But I'm confused on why $\pi(S) = (S + I)/I$ and not $\pi(S) = S/I$.