Embedding dimension

Remarks

Since $\mathfrak m$ is finitely generated, Nakayama's lemma shows that $\operatorname{edim} R$ is the minimal number of generators of the ideal $\mathfrak m$. The dimension theorem for local rings gives $\dim R \le \operatorname{edim} R$, where $\dim R$ is the Krull dimension. When equality holds, $R$ is a regular local ring.

Example: for a field $k$, the ring $k[\![x_1, \ldots, x_n]\!]$ has $\mathfrak m = (x_1, \ldots, x_n)$, and the classes of the $x_i$ form a basis of $\mathfrak m/\mathfrak m^2$, so $\operatorname{edim} = n$. For the cusp $R = k[x, y]_{(x,y)}/(y^2 - x^3)$, one has $\operatorname{edim} R = 2$ but $\dim R = 1$.

Non-example: without the Noetherian hypothesis, $\dim_k \mathfrak m / \mathfrak m^2$ need not count generators of $\mathfrak m$. In a valuation ring with value group $\Q$, every element of positive value $q$ is a product of two elements of value $q/2$. So $\mathfrak m = \mathfrak m^2$ and $\mathfrak m / \mathfrak m^2 = 0$, although $\mathfrak m \ne 0$; here $\mathfrak m$ is not finitely generated.