The results in this paper expand the fundamental decomposition theory of ideals pioneered by Emmy Noether. Specifically, let
I
I
be an ideal in a local ring
(
R
,
M
)
(R,M)
that has
M
M
as an embedded prime divisor, and for a prime divisor
P
P
of
I
I
let
I
C
P
(
I
)
I{C_P}(I)
be the set of irreducible components
q
q
of
I
I
that are
P
P
-primary (so there exists a decomposition of
I
I
as an irredundant finite intersection of irreducible ideals that has
q
q
as a factor). Then the main results show: (a)
I
C
M
(
I
)
=
∪
{
I
C
M
(
Q
)
;
Q
is a
MEC
of
I
}
I{C_M}(I) = \cup \{ I{C_M}(Q);Q\;{\text {is a }}\operatorname {MEC} {\text { of }}I\}
(
Q
Q
is a MEC of
I
I
in case
Q
Q
is maximal in the set of
M
M
-primary components of
I
I
); (b) if
I
=
∩
{
q
i
;
i
=
1
,
…
,
n
}
I = \cap \{ {q_i};i = 1, \ldots ,n\}
is an irredundant irreducible decomposition of
I
I
such that
q
i
{q_i}
is
M
M
-primary if and only if
i
=
1
,
…
,
k
>
n
i = 1, \ldots ,k > n
, then
∩
{
q
i
;
i
=
1
,
…
,
k
}
\cap \{ {q_i};i = 1, \ldots ,k\}
is an irredundant irreducible decomposition of a MEC of
I
I
, and, conversely, if
Q
Q
is a MEC of
I
I
and if
∩
{
Q
j
;
j
=
1
,
…
,
m
}
\cap \{ {Q_j};j = 1, \ldots ,m\}
(resp.,
∩
{
q
i
;
i
=
1
,
…
,
n
}
\cap \{ {q_i};i = 1, \ldots ,n\}
) is an irredundant irreducible decomposition of
Q
Q
(resp.,
I
I
) such that
q
1
,
…
,
q
k
{q_1}, \ldots ,{q_k}
are the
M
M
-primary ideals in
{
q
1
,
…
,
q
n
}
\{ {q_1}, \ldots ,{q_n}\}
, then
m
=
k
m = k
and
(
∩
{
q
i
;
i
=
k
+
1
,
…
,
n
}
)
∩
(
∩
{
Q
j
;
j
=
1
,
…
,
m
}
)
( \cap \{ {q_i};i = k + 1, \ldots ,n\} ) \cap ( \cap \{ {Q_j};j = 1, \ldots ,m\} )
is an irredundant irreducible decomposition of
I
I
; (c)
I
C
M
(
I
)
=
{
q
,
q
is maximal in the set of ideals that contain
I
and do not contain
I
:
M
}
I{C_M}(I) = \{ q,q\;{\text {is maximal in the set of ideals that contain }}I\;{\text {and do not contain }}I:M\}
; (d) if
Q
Q
is a MEC of
I
I
, then
I
C
M
(
Q
)
=
{
q
;
Q
⊆
q
∈
I
C
M
(
I
)
}
I{C_M}(Q) = \{ q;Q \subseteq q \in I{C_M}(I)\}
; (e) if
J
J
is an ideal that lies between
I
I
and an ideal
Q
∈
I
C
M
(
I
)
Q \in I{C_M}(I)
, then
J
=
∩
{
q
;
J
⊆
q
∈
I
C
M
(
I
)
}
J = \cap \{ q;J \subseteq q \in I{C_M}(I)\}
; and, (f) there are no containment relations among the ideals in
∪
{
I
C
P
(
I
)
\cup \{ I{C_P}(I)
;
P
P
is a prime divisor of
I
I
}.