Ideal Contains Extension of Contraction

Theorem
Let $A$ and $B$ be commutative rings with unity.

Let $f : A \to B$ be a ring homomorphism.

Let $\mathfrak b \subseteq B$ be an ideal.

Then $\mathfrak b$ contains the extension of its contraction by $f$:
 * $\mathfrak b^{ce} \subseteq \mathfrak b$

Proof
Since $\mathfrak b^{ce}$ is generated by $f \sqbrk {\mathfrak b ^c}$ according to, it is contained in $\mathfrak b$