Let $\frak G$ be a Lie algebra and let $M$ be a $\frak G$-module generated by a vector $v$ satisfying some set of defining relations denoted by $R$. I mean, $M = U(\frak G)/\langle R \rangle$, where $\langle R \rangle$ is the $U(\frak G)$-submodule generated by $R$ and $U(\frak G)$ denotes the universal enveloping algebra of $\frak G$. In this case, $v = \overline 1$.

Suppose that $\frak B \subset \frak G$ is an ideal such that ${\frak B} M = 0$. So, we can naturally regard $M$ as a $\frak G/\frak B$-module.

Can we say that $M$ is a $\frak G/\frak B$-module given by one generator and relations coming from the defining relations of $M$? How to formalize this fact?

I feel that it is some sort of consequence of the Theorem of Isomorphisms, but I don't know how to write it in a formal way.