Let $R$ be a ring. Say an ideal $\frak p$ is prime if $IJ\subset \mathfrak p\implies I\subset \mathfrak p\text{ or }J\subset \mathfrak p$.
I need to prove that this implies the stronger condition in which $I,J$ are assumed both right ideals, but I don't see how to do this.
If $I$ and $J$ are right ideals, $RJ$ and $RI$ are ideals, and $IRJ\subseteq IJ\subseteq P$. Since $P$ is two sided, $RIRJ\subseteq P$ also. By hypothesis, $RI$ or $RJ$ is contained in $P$, whence $I$ or $J$ is contained in $P$.