# Is “P is a left primitive ideal” implies that there is a left maximal ideal…?

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.

• Did you mean "..., then $A\subseteq P$ "? – Rasmus Oct 29 '11 at 12:25
• @Rasmus: As it's told to me, $A$ is a proper one. – Basil R Oct 29 '11 at 16:05
• The claim cannot hold as written: if we take $A=P$, then $A\subsetneq L$, but $A$ is clearly not properly contained in $P$. – Arturo Magidin Oct 29 '11 at 22:00
• Also, do you mean this to be a definition of primitive, or do you mean it to be a sufficient condition for primitivity? – Arturo Magidin Oct 29 '11 at 22:24
• @Basil: I would suggest correcting it, and then posting the reference/proof you found as an answer. – Arturo Magidin Oct 30 '11 at 18:24