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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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There is no strong relations between $J_A$ and $J_B$ if you don't have some condition on $A\to B$.

First, it is not true that $f(J_A)\subseteq J_B$: just take $A$ integrally closed and $B$ not integrally closed. Then $1\in J_A$ but $1=f(1)\notin J_B$.

Second, suppose $B$ is the integral closure of $A$ and $B\ne A$. Then $J_B=B$ and $J_B$ is not contained in $f(J_A)B$.

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