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For Rings $X, Y$ with a ring homomorphism $f: X \rightarrow Y$. Let $\mathfrak{a}_1, \mathfrak{a_2} \subseteq X, \mathfrak{b}_1, \;\mathfrak{b_2} \subseteq Y$ be ideals.

I was able to prove:

$$\mathfrak{b}_1^c + \mathfrak{b_2}^c \subseteq (\mathfrak{b}_1 + \mathfrak{b}_2)^c$$

and

$$(\mathfrak{a}_1 \cup \mathfrak{a}_2)^e \subseteq \mathfrak{a}_1^e \cup \mathfrak{a}_2^e$$

but fail to prove the other inclusion. Do these inclusions even hold? And how can I see this?

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To see that we don't have equality for sums of contractions, consider the ring homomorphism $K\to K\times K$ and take the ideals $\mathfrak b_1=K\times0$ and $\mathfrak b_2=0\times K$. Then $\mathfrak b_i^c=0$, but $\mathfrak b_1+\mathfrak b_2=K\times K$. The result is true, however, if $X\to Y$ is onto.

To see that we don't have equality for intersections of extensions, consider the ring homomorphism $K[x,y]\to K[t]$, $x,y\mapsto t$, and take the ideals $\mathfrak a_1=(x)$ and $\mathfrak a_2=(y)$. Then $\mathfrak a_i^e=(t)$, but $(\mathfrak a_1\cap\mathfrak a_2)^e=(xy)^e=(t^2)$.

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