# Sum and Intersection of Contraction and Extension

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?

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 $$(\mathfrak a_1\cap\mathfrak a_2)^e=(xy)^e=(t^2)$$.