For finitely generated modules
N
⊊
M
N \subsetneq M
over a Noetherian ring
R
R
, we study the following properties about primary decomposition: (1) The Compatibility property, which says that if
Ass
(
M
/
N
)
=
{
P
1
,
P
2
,
…
,
P
s
}
\operatorname {Ass} (M/N)=\{ P_1, P_2, \dots , P_s\}
and
Q
i
Q_i
is a
P
i
P_i
-primary component of
N
⊊
M
N \subsetneq M
for each
i
=
1
,
2
,
…
,
s
i=1,2,\dots ,s
, then
N
=
Q
1
∩
Q
2
∩
⋯
∩
Q
s
N =Q_1 \cap Q_2 \cap \cdots \cap Q_s
; (2) For a given subset
X
=
{
P
1
,
P
2
,
…
,
P
r
}
⊆
Ass
(
M
/
N
)
X=\{ P_1, P_2, \dots , P_r \} \subseteq \operatorname {Ass}(M/N)
,
X
X
is an open subset of
Ass
(
M
/
N
)
\operatorname {Ass}(M/N)
if and only if the intersections
Q
1
∩
Q
2
∩
⋯
∩
Q
r
=
Q
1
′
∩
Q
2
′
∩
⋯
∩
Q
r
′
Q_1 \cap Q_2\cap \cdots \cap Q_r= Q_1’ \cap Q_2’ \cap \cdots \cap Q_r’
for all possible
P
i
P_i
-primary components
Q
i
Q_i
and
Q
i
′
Q_i’
of
N
⊊
M
N\subsetneq M
; (3) A new proof of the ‘Linear Growth’ property, which says that for any fixed ideals
I
1
,
I
2
,
…
,
I
t
I_1, I_2, \dots , I_t
of
R
R
there exists a
k
∈
N
k \in \mathbb N
such that for any
n
1
,
n
2
,
…
,
n
t
∈
N
n_1, n_2, \dots , n_t \in \mathbb N
there exists a primary decomposition of
I
1
n
1
I
2
n
2
⋯
I
t
n
t
M
⊂
M
I_1^{n_1}I_2^{n_2}\cdots I_t^{n_t}M \subset M
such that every
P
P
-primary component
Q
Q
of that primary decomposition contains
P
k
(
n
1
+
n
2
+
⋯
+
n
t
)
M
P^{k(n_1+n_2+\cdots +n_t)}M
.