Prime ideals definition

Question

If $A,~B$ are two ideals of a ring $R$ and $P$ is a prime ideal of $R$ such that $A\subseteq P$ or $B\subseteq P$ . Does this implies to $AB\subseteq P$? i.e. Does the result true since $P$ is prime?

Answer 1

If $A \subset P$ or $B \subset P$ then $$ AB \subset A \subset P \quad\text{ or } \quad AB \subset B \subset P. $$ The ideal $P$ does not need to be prime.

Answer 2

By hypothesis, $A \subset P$ or $B \subset P$. Without loss of generality, we can assume that $A \subset P$. Let $x = ab \in AB$. As $a \in P$ and $b \in B \subset R$ and $P$ is a ideal of $R$, we have $ab \in P$. Actually, P does not have to be a prime ideal of $R$.