Suppose I want to show that $\mathbb{C}[X]$ is the unique simple $A_1$-submodule of $\mathbb{C}[X,X^{-1}]$, where $A_1 = A_1(\mathbb{C})$ is the first Weyl Algebra.
It is not difficult to show it is simple. Moreover, I read that if $\mathbb{C}[X] \subseteq A_1 * \alpha$, for any $0 \neq \alpha \in \mathbb{C}[X,X^{-1}] / \mathbb{C}[X]$, then uniqueness follows. Although I am finding it difficult to understand why this is the case. Any suggestions?
For any $f\in \mathbb{C}[X,X^{-1}]$, there is some $n\in\mathbb{N}$ such that $X^nf\in \mathbb{C}[X]$. So, if $f$ is nonzero, the $A_1$-submodule generated by $f$ contains a nonzero element of $\mathbb{C}[X]$, and thus contains all of $\mathbb{C}[X]$ since $\mathbb{C}[X]$ is simple. So, every nonzero submodule of $\mathbb{C}[X,X^{-1}]$ contains $\mathbb{C}[X]$, and thus such a submodule cannot be simple unless it is equal to $\mathbb{C}[X]$.