The Noether-Lasker theorem states that for a commutative Noetherian ring 
and a nonzero left unital
R-module 
that is a finitely
generated module, there is a finite set of submodules 
with the following
properties (Altman and Kleiman 2013, pp. 109-110).
1. 
,
and 
is not contained in 
for all 
.
2. Each quotient module 
is nonzero and 
-primary for some prime ideal 
of 
.
3. The prime ideals 
are all distinct for 
.
4. If 
does not contain 
for any 
, then 
is uniquely determined by 
and 
.
Here, 
-primary
means that every 
acts on 
either injectively or with
some positive integer power
acting as zero, and 
is exactly the set of 
satisfying the latter condition (Milne 2014, p. 90).
