By definition, a primitive ideal $P$ exists if there is a simple $R$-module $S$ such that $Ann(S)$=$P$. I saw another statement as follows:
"$P$ is a primitive ideal of a ring if there is a left maximal ideal $L$ such that $P \subsetneq L \ $ and for any ideal $A$ of $R$, $A \subsetneq L\ $, then $A\subseteq P$ "
If this claim is right please note me some good references. Thanks.