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Let $A$ be a Noetherian ring, $\mathfrak{p}\subset A$ a prime ideal of height $p$, $N$ an $A_{\mathfrak{p}}$-module of finite length, $M,M'\subset N$ finitely generated $A$-submodules such that $M\varsubsetneq M'$ and $M_{\mathfrak{p}}=M'_{\mathfrak{p}}=N$. Then is it true that every minimal associated prime of $M'/M$ has height $p+1$? I could show that they must have height $\geq p+1$.

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No. If $A = k[x,y]$ is the polynomial ring in two variables, $\mathfrak{p}$ is the zero ideal, $N = A_{\mathfrak{p}} = k(x,y)$ is the field of fractions of $A$, $M := (x,y) \subsetneq M' := A \subseteq N$ all satisfy your conditions.

But the only associated prime of $M' / M$ is $(x,y)$ which has height $2$.

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