I've seen two definitions of Krull dimension of a module $M$ over a (commutative) ring $R$, and their equivalence does not seem obvious:
Matsumura on page 31 of his book Commutative Ring Theory defines it as
$\dim M=\dim R/\operatorname{Ann}(M)=$ maximal length of a chain of primes in $V(\operatorname{Ann}(M)).$
Enochs and Jenda on page 54 of Relative Homological Algebra define it as
$\dim M=\dim {\rm Supp}(M)=$ maximal length of a chain of primes in ${\rm Supp}(M).$
I guess this "maximal length" is the same for two sets above, but what's the proof? Otherwise how are two definitions equivalent?
PS: I already know that $\mathrm{Supp}(M)\subseteq V(\operatorname{Ann}(M))$ and that both definitions are equivalent for finitely generated modules.
These definitions are not the same in general, if $M$ is not f.g.
Consider the module $\mathbb Q_p/\mathbb Z_p$ over $\mathbb Z_p$. Its annihilator is $0$, so the first definition gives dimension $1$. On the other hand, its support is the closed point of Spec $\mathbb Z_p$, and so the second definition gives dimension $0$.
If you are reading an article that applies the notion of dimension in the non-f.g. context, then you will either have to look and see if the author defines their terms, or else determine from the context (e.g. how they argue) which definition is in use.