Ulrich module

A maximal Cohen-Macaulay module $M$ over a Noetherian local ring $(R,𝔪,k)$ is Ulrich if $e(M)=\mu (M)$, where $e(M)$ is the Hilbert-Samuel multiplicity of $M$ and $\mu (M)$ is the minimal number of generators of $M$. When $M$ is a maximal Cohen-Macaulay module and $𝔪$ has a minimal reduction $I$ generated by a system of parameters, $M$ is Ulrich if and only if $𝔪M=IM$.