Rank of a module when the base ring is not a domain

Question

Suppose $R$ is a commutative Noetherian local ring with $1$, which is not a domain. Let $M$ be a (non-free) finite $R$-module. What is meant by rank of $M$ in this case?

Answer 1

The paper that I am reading assumes rank of a module always exists. Hence, it may be using a different definition of rank, which may (or may not) coincide with the definition provided in the previous answer, when $M\otimes_RQ$ is free over $Q$. After I posted my question I found the following definition of rank in the book Syzygies (By E. Graham Evans, Phillip Griffith) on page 2:

Rank of $M$ = maximum of {rank of $M/\mathfrak{p}M$ over $R/\mathfrak{p}$, where $\mathfrak{p}$ runs over the set of minimal prime ideals of $R$}.