Let $k$ be a field, $A$ a finitely generated local integral domain over $k$ of Krull dimension $1$, and $A'$ be the normalization of $A$ in the fraction field of $A$. Then is $A'/A$ a torsion $A$-module, i.e., $A' /A \cong \bigoplus A/g_i $ for some $g_i \in A$?
I want to use it to show that $\dim_k A'/A$ is finite and the equation $\dim_kA'/fA' = \dim_k A/fA$, where $f$ is a non-unit element of $A$.
I've shown that $A'$ is a finite $A$-module.
Thank you very much.