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Let $f : A \to B$ be a homomorphism of finitely generated $k$-algebras, where $k$ is a field. Let $J_A$ and $J_B$ denote the conductor ideals of $A$ and $B$ respectively for the corresponding normalizations in the quotient fields (assume that $A, B$ are domains). Is it true that $f(J_A) \subseteq J_B$ ? What is the relationship between $f^{-1}(J_B)$ and $J_A$ ?

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