I am studying ideal extension and have the following question that I cannot answer. Please help me.
Let $A$ be a commutative ring with unity and $B$ is a subring of $A$ containing the unity of $A$. Suppose also that $A$ is neither a ring of fractions of $B$ nor a field. Let $P \neq \{0\}$ be a prime ideal of $B$ and $P'$ be the extension of $P$ in $A$, that is, $P' = \{\sum_{\text{finite sum}} a_{i}p_{i} \mid a_{i} \in A \text{ and } p_{i} \in P\}$. What is a necessary and sufficient condition so that $P' \neq A$?