Equivalence of definitions of Krull dimension of a module

Equivalence of definitions of Krull dimension of a module

I've seen two definitions of Krull dimension of a module $M$ over a
(commutative) ring $R$, and their equivlance 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.