This is came from the definition of system of parameters in Stanley's 1996 book "Combinatorics and Commutative Algebra." In this book at page 33, he defines the (homogeneous) system of parameters $\theta_{1},\dots, \theta_{d}$ of a $\mathbb{Z}^{m}$-graded $R$-module $M$ (where $R$ is $\mathbb{N}^{n}$-graded $k$-algebra over some field $k$) as
$$\dim(M/(\theta_{1}M+\dots +\theta_{d}M))=0.$$
However, the standard definition (especially, in Matsumura's famous book) of a system of parameters is a set of elements such that the length of the module $M/(\theta_{1}M+\dots \theta_{d}M)$ is finite.
Now he claims that a system of parameter exists if and only if $M$ is finitely generated $k[\theta_{1},\dots, \theta_{d}]$-module. However, this formula only holds when Krull dimension 0 implies finite length.
Does this (Krull dimension 0 module is of finite length) true? Or is there any counter example? I think it has a counter example when $M$ is infinitely generated module, but I'm not sure at this point. Any hints will be appreciated.