Multiplicity of a local ring

Remarks

Example: let $R = k[x, y]_{(x, y)}/(y^2 - x^3)$, with $k$ a field. For $n \ge 2$, $\ell(R/\mathfrak m^n) = 2n - 1$, since the monomials of degree $< n$ in $k[x, y]$ number $n(n+1)/2$ and those divisible by $y^2$ number $(n-2)(n-1)/2$. So $d = 1$, the leading coefficient is $2 = e/1!$, and $e(R) = 2$. Non-example: the function $n \mapsto \ell(R/I^n)$ for an ideal $I$ that is not $\mathfrak m$-primary, such as $I = (y)$ in $k[x, y]_{(x, y)}$, takes infinite values, so it has no multiplicity in this sense. A regular local ring has multiplicity $1$, and an Artinian local ring $R$ has $e(R) = \ell(R)$.