I am reading Introduction to Commutative Algebra / Atiyah & Macdonald, Theorem 5.11 ("Going-up theorem").
The statement is:
Let $A \subset B$ be rings, $B$ integral over $A$; let $p_1 \subset \dotsm \subset p_n$ be a chain of prime ideals of $A$ and $q_1 \subset \dotsm \subset q_m$ (m < n) a chain of prime ideals of $B$ such that $q_i \cap A = p_i$ ($1 \leq i \leq m$). Then the chain $q_1 \subset \dotsm \subset q_m$ can be extended to a chain $q_1 \subset \dotsm \subset q_n$ such that $q_i \cap A = p_i$ for $1 \leq i \leq n$.
I read the proof and I don't think the fact that $q_1$ (and hence $p_1$) is prime is used.
My question:
Does the conclusion still hold if we omit the assumption that $q_1$ (and $p_1$) is prime?