Idempotents and cyclic codes Let $C_1 = \langle e_1(x) \rangle$, $C_2 = \langle e_2(x) \rangle$ cyclic codes, where $e_1(x)$ and $e_2(x)$ are idempotents.
I know what cyclic codes and idempotents are, but why can one deduce the following: $C_1 \subset C_2  \Leftrightarrow  e_1(x) e_2(x) = e_1(x)$?

I believe you are talking about an idempotent $e=e(x)\in F[x]/(x^n-1)$.  The fact you are speaking of is actually true for idempotents in any commutative ring with identity.
Suppose $e,f$ are two idempotents in a commutative ring $R$.
Then it is elementary to show that $(e)\cap (1-e)=\{0\}$ and $(f)\cap (1-f)=\{0\}$.
Now if $(e)\subseteq (f)$, then $e-ef=e(1-f)\in (f)\cap (1-f)=\{0\}$. Therefore $e=ef$.
In the other direction, suppose $e=ef$: then clearly $e\in (f)$ and $(e)\subseteq (f)$.
