Currently I'm working on my final project, and while I read some literature about my project, I found this expressions:
Given $f: R\to R'$ is a ring homomorphism. For any $y_1,y_2\in R'$, $$ \{x \mid x\in f^{-1}(y_1-y_2)\} \supseteq \{x_1-x_2 \mid x_1 \in f^{-1}(y_1),\, x_2\in f^{-1}(y_2)\}$$
How can I explain that this expression is true? I want to try using homomorphism properties, but I don't think that those properties are always true on $f^{-1}$ because the homomorphism isn't always injective. And to proving superset, I know it means that every element of RHS must be element of LHS.
Fix $y_1, y_2 \in R'$ and take some $x_1 - x_2 \in R$ from the set on the right-hand side. So $f(x_1) = y_1$ and $f(x_2) = y_2$. Now, since this is a ring homomorphism, $$ f(x_1 - x_2) = f(x_1) - f(x_2) = y_1 - y_2, $$ which shows that $x = x_1 - x_2$ is an element of the set on the left-hand side.
What's interesting is that the converse only holds in certain situations. What condition on $f$ allows for the reverse set containment?