How to prove that the set of maximal elements of a set of prime ideals is finite Let $A$ be a subset of ${\rm Spec}(R)$ with $R$ noetherian

Are there any techniques to prove that ${\rm max}(A)$ (ie the set of maximal elements of $A$) is finite?

I'm looking for equivalent conditions or conditions that imply this finiteness.

Particularly I'm thinking on the case $A:={\rm Ass}_R(M)$
there are different ways:
Matsumura:
from   



and   



Concludes    





Sharp in "Steps in Commutative Algebra" uses minimal primary
decomposition of modules (see 9.25 - 9.33)
there are different ways:
Matsumura:
from   



and   



Concludes    





Sharp in "Steps in Commutative Algebra" uses minimal primary
decomposition of modules (see 9.25 - 9.33)