TITLE: Equivalence of definitions of Krull dimension of a module
QUESTION [5 upvotes]: 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.

REPLY [2 votes]: Another example (of non-finitely generated module) showing the two definitions of dimension give different results is the $\mathbb Z$-module $M=\oplus_{n\ge1}\mathbb Z/2^n\mathbb Z$. (Maybe there are users who find hardly understood MattE's answer using the $p$-adics.)
We have $\operatorname{Ann}M=(0)$, and therefore $\dim M=1$, if one uses the first definition of dimension.
On the other side, $\operatorname{Supp}M=\{2\mathbb Z\}$, and then $\dim M=0$, if one uses the second definition.