Let $m$ be a maximal ideal of a commutative ring $R$ with $1$. Can we construct a generating set $\{x_i\}_{i\in I}$ for the injective envelope $E(R/m) $ of $R/m$ such that $R/m\not\subseteq\langle x_i\rangle$ for each $i\in I$?
Or is there description for a set of generation set of $E(R/m) $ ?
$R/\mathfrak m$ is simple, so $\langle x_i \rangle \cap R/\mathfrak{m}$ is either $0$, or $R/\mathfrak{m}$. But $R/\mathfrak{m}$ is an essential submodule, so the former cannot happen, assuming that $x_i \neq 0$. That is, given any set $\{x_i\}_i$ of nonzero generators of $E(R/\mathfrak{m})$, we have $R/\mathfrak{m} \subseteq \langle x_i \rangle$ for each $x_i$.
