I have not found this question on the website, maybe because it's easy, so if it has been asked before I will delete my question.
Let $A,B$ be two rings, $\mathfrak{a}$ an ideal of $A$ and $\mathfrak{b}$ is an ideal of $B$. Lastly, let $f: A \rightarrow B$ be a ring homomorphism. How can I show the following properties?
$\mathfrak{b}^{ce} \subset \mathfrak{b}$.
$\mathfrak{a} \subset \mathfrak{a}^{ec}$.
The second seems trivial, since $\mathfrak{a}^e=B\mathfrak{a}$ $ \implies f^{-1}(B \mathfrak{a}) =\mathfrak{a}^{ec}= \ker f\mathfrak{a} \supset \mathfrak{a} $.
However, how can I show the first inclusion? I think it does not look "natural".
Hint: use the fact that $f(f^{-1}(\mathfrak b))\subseteq \mathfrak b$.