Primitive ideals of an enveloping algebra

Let $\mathfrak{g}$ be a complex Lie algebra. A two-sided ideal $I⊴U(\mathfrak{g})$ (Left, right and two-sided ideals) is primitive if there exists a simple left $U(\mathfrak{g})$-module $M$ (Simple module: a nonzero module with no proper nonzero submodule) with

$I={\mathrm{Ann}}_{U(\mathfrak{g})}(M)=\{u\in U(\mathfrak{g}):um=0\text{ for every }m\in M\}$

(The annihilator of a module over an enveloping algebra). Equivalently: a two-sided ideal is primitive if it is the kernel of the action of $U(\mathfrak{g})$ on some simple module. Every primitive ideal is proper, and $U(\mathfrak{g})\mathrm{/}I$ admits the faithful simple module $M$; no highest-weight hypothesis is imposed on $M$.

• Properness. The module $M$ is nonzero, and $1\in U(\mathfrak{g})$ acts as the identity, so $1\mathrm{\notin }{\mathrm{Ann}}_{U(\mathfrak{g})}(M)$; a two-sided ideal containing $1$ is all of $U(\mathfrak{g})$, so a primitive ideal is a proper ideal. This is the only place nonzero-ness of $M$ is used in the definitional consequences.
• Faithfulness after quotienting. By The annihilator of a module over an enveloping algebra the action of $U(\mathfrak{g})\mathrm{/}I$ on $M$ is faithful, so the equivalent kernel formulation and the quotient statement describe the same situation.
• No highest-weight hypothesis. The simple module $M$ in the definition is arbitrary; in particular a primitive ideal need not be realised by a highest weight module in the definition itself. For finite-dimensional complex semisimple $\mathfrak{g}$, realization by a simple highest-weight module is Duflo's theorem, not part of this definition.