embedding dimension is minimal maximal ideal generator number

Given: The objects and hypotheses in the statement. We work with the Axiom of Choice; cited dependent-choice and resolution-existence hypotheses are retained.

embedding dimension and regular local ring: For a nonzero commutative Noetherian local ring $(R,\mathfrak{m},k)$, define $\mathrm{edim}R={\dim }_k(\mathfrak{m}\mathrm{/}{\mathfrak{m}}^2)$. The ring is regular local when $\mathrm{edim}R=\dim R$. The cotangent space is intrinsic, and is finite-dimensional because $\mathfrak{m}$ is finitely generated.

Assuming the Axiom of Choice, minimal generators over a local ring are exactly residue-field bases: Assume the Axiom of Choice. Let $(R,\mathfrak{m})$ be a local ring with residue field $k=R\mathrm{/}\mathfrak{m}$, and let $M$ be a finitely generated left $R$-module. A finite generating set $x_1,\dots ,x_r$ of $M$ is minimal if and only if the images of $x_1,\dots ,x_r$ in $M\mathrm{/}\mathfrak{m}M$ form a $k$-basis. In particular every minimal generating set of $M$ has the same cardinality.

Assuming the Axiom of Choice, generators modulo an ideal in the Jacobson radical lift to generators: Assume the Axiom of Choice. Let $R$ be a commutative ring, let $I⊴R$ satisfy $I\subseteq J(R)$, and let $M$ be a finitely generated left $R$-module. If elements $x_1,\dots ,x_r\in M$ generate $M\mathrm{/}IM$, then $x_1,\dots ,x_r$ generate $M$.