I have been reading the following paper: https://www.sciencedirect.com/science/article/pii/S002240491000263X

Proposition 2.4(ii) shows that if $\mathfrak D$ is a ring of $k$-linear differential operators of the polynomial ring $R=k[x_1, \cdots , x_n]$, then for every $k$-rational maximal ideal $\mathfrak m$ of $R$ (i.e. a maximal ideal $\mathfrak m \vartriangleleft R$ for which the canonical map $k \rightarrow R/\mathfrak m$ is an isomorphism), the socle of $\mathfrak D/\mathfrak D \mathfrak m$ is generated by $\overline 1$. There is a part in the proof in this proposition which reads that in order to demonstrate this assertion, "It remains to show that every non-zero element $z$ can be sent to $1$ by an element of $R$."

I am not sure why at that point in the proof, this is sufficient to show the desired assertion on the socle of $\mathfrak D/\mathfrak D \mathfrak m$. The only definitions of socle in this (non-local) scenario that I am aware of are that the socle is the sum of simple submodules or equivalently the intersection of all essential submodules (and that is basically all I know about the socle, so I apologize for the possible naivete of the question) and I can't seem to be able to correlate what either of these two definitions has to do with the sufficiency claimed above. I would really appreciate any help or reference here, thank you.



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