Does the least number of generators of the parameter ideal and degree of Samuel function depend only on the annihilator

Question

Let $R$ be a local Noetherian ring, and $M$ is a finitely generated module over $R$. Set $\delta(M)$ to be the least number of generators of a parameter ideal of $M$ (a parameter ideal $I$ of $M$ by definition satisfy $l(M/IM) < \infty$). Let $d(M)$ denote the degree of the Samuel function of $M$ with respect to $I$-adic filtration.

Is it true that $\delta(M)=\delta(R/\operatorname{Ann}(M))$ and $d(M)=d(R/\operatorname{Ann}(M))$?

Answer 1

The result follows immediately from Theorem 13.4 in Matsumura's Commutative Ring Theory and from the fact that $$\text{dim}(M)=\text{dim}(M/\text{Ann}(M)),$$ by definition of Krull dimension of a module.