On the embedded primary components of ideals. IV
Authors: Heinzer William , Ratliff L. J. , Shah Kishor 
Published: June 2012

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)
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I upper C Subscript upper M Baseline left-parenthesis upper I right-parenthesis equals union left-brace upper I upper C Subscript upper M Baseline left-parenthesis upper Q right-parenthesis semicolon upper Q is a upper M upper E upper C of upper I right-brace">

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
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis intersection left-brace q Subscript i Baseline semicolon i equals k plus 1 comma ellipsis comma n right-brace right-parenthesis intersection left-parenthesis intersection left-brace upper Q Subscript j Baseline semicolon j equals 1 comma ellipsis comma m right-brace right-parenthesis">

(
∩
{

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
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I upper C Subscript upper M Baseline left-parenthesis upper Q right-parenthesis equals left-brace q semicolon upper Q subset-of-or-equal-to q element-of upper I upper C Subscript upper M Baseline left-parenthesis upper I right-parenthesis right-brace">

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

}.