The annihilator of a module over an enveloping algebra

Let $\mathfrak{g}$ be a complex Lie algebra and let $M$ be a nonzero left $U(\mathfrak{g})$-module with action ${\rho }_M\text{\hspace{0.17em}⁣}:U(\mathfrak{g})\to {\mathrm{End}}_{\mathbb{C}}(M)$, the unital extension of the $\mathfrak{g}$-action supplied by Lie algebra actions extend to unital actions of the enveloping algebra. The annihilator of $M$ is

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

and for $m\in M$ one writes $\mathrm{Ann}(m)=\{u\in U(\mathfrak{g}):um=0\}$. Then ${\mathrm{Ann}}_{U(\mathfrak{g})}(M)$ is a two-sided ideal of $U(\mathfrak{g})$ (Left, right and two-sided ideals), equal to the intersection of the left ideals $\mathrm{Ann}(m)$, $m\in M$; the action descends to a faithful action of the quotient algebra $U(\mathfrak{g})\mathrm{/}{\mathrm{Ann}}_{U(\mathfrak{g})}(M)$ on $M$.

• The annihilator is the kernel of the action. ${\rho }_M$ is a $\mathbb{C}$-algebra homomorphism, so its kernel is a two-sided ideal by The kernel of a ring homomorphism is a two-sided ideal; unwinding definitions, this kernel is exactly the set of $u$ annihilating every $m\in M$.
• Pointwise annihilators are left ideals. For fixed $m$, the map $u\mapsto um$ is $\mathbb{C}$-linear, so $\mathrm{Ann}(m)$ is an additive subgroup; and $u\in \mathrm{Ann}(m)$ implies $vu\in \mathrm{Ann}(m)$ for every $v\in U(\mathfrak{g})$ because $(vu)m=v(um)=0$. Thus each $\mathrm{Ann}(m)$ is a left ideal, and ${\mathrm{Ann}}_{U(\mathfrak{g})}(M)={\bigcap }_{m\in M}\mathrm{Ann}(m)$ because a $u$ annihilating every $m$ is exactly one lying in every pointwise annihilator.
• Faithfulness of the quotient action. If $u+\mathrm{Ann}M$ acts as zero on $M$, then $um=0$ for all $m\in M$, so $u\in \mathrm{Ann}M$ and the class is zero. The quotient therefore acts faithfully, and $M$ is a left module over it by the same formula (Unital left and right modules over a ring; unqualified module means left module).