%From ratliff@math.ucr.edu Fri Nov 11 10:19:39 1994

\magnification=\magstephalf
\input amstex
\documentstyle{amsppt}
\hoffset .25 true in
\NoBlackBoxes
\def\CE{corner-element\ }
\def\CES{corner-elements\ }
\def\R{\text{Rad}}
\topmatter
\title PARAMETRIC DECOMPOSITION OF MONOMIAL IDEALS (I)
\endtitle
\author
William Heinzer,  L. J. Ratliff, Jr. and Kishor Shah
\endauthor
\thanks{The first author's research on this paper was supported in part by the National Science Foundation, Grant DMS-9101176.}
\endthanks
\address
\break
{Department of Mathematics, Purdue University, W. Lafayette, IN 47907}\break
{Department of Mathematics, University of California, Riverside, CA 92521}\break
{Department of Mathematics, Southwest Missouri State University, Springfield, MO 65804}
\endaddress
\subjclass AMS (MOS) Subject Classification Numbers:  Primary: 13A17, 13C99.
Secondary: 13B99, 13H99\endsubjclass
\abstract
Emmy Noether showed that every ideal in a Noetherian ring admits a decomposition
into irreducible ideals.  In this paper we explicitly calculate this
decomposition in a fundamental case.  Specifically, let  $R$  be a commutative
ring with identity, let  $x_1,\dots,x_d$ $(d>1)$  be an  $R$ -sequence,
let  $X=(x_1,\dots,x_d)R$, and let  $I$  be a monomial ideal
(that is, a proper ideal
generated by monomials  $x_1^{e_1}\cdots x_d^{e_d})$  such
that  $Rad(I)$ $=$ $Rad(X)$.  Then the main result gives a canonical and
unique decomposition of  $I$  as an irredundant finite intersection of ideals of
the form  $(x_1^{n_1},\dots,x_d^{n_d})R$, where the
exponents  $n_1,\dots,n_d$  are
positive integers.  Specifically, if  $z_1,\dots,z_m$  are the monomials
in  $(I:X)-I$, and if  $z_j=x_1^{a_{j,1}-1}\cdots x_d^{a_{j,d}-1}$,
then  $I$ $=$ $\cap\{(x_1^{a_{j,1}},\dots,x_d^{a_{j,d}})R$; $j$ $=$
$1,\dots,m\}$.  We also calculate the decomposition of
the ideals  $I^{[k]}$  generated by the  $k$ -th powers of the monomial
generators of  $I$.
The methods we use are algebraic, but they were suggested by the geometry
of lattices.
\endabstract

\endtopmatter
\baselineskip 18pt
\noindent
{\smc{\bf 1.  Introduction}}.  {\it Throughout this paper}, $R$ is a
commutative ring with identity  $1\not= 0$, $x_1,\dots, x_d$ $(d>1)$  is
an  $R$ -sequence, $X=(x_1,\dots,x_d)R$, and  $I$  is a monomial ideal (that
is, a proper ideal generated by monomials  $x_1^{e_1}\cdots x_d^{e_d})$  such
that  $Rad(I)$ $=$ $Rad(X)$.

It is known (for example, see [HRS2, (3.15)]) that in a regular local
ring  $R$  of altitude two, irreducible ideals are parameter ideals.
Therefore in altitude two regular local rings, Emmy Noether's fundamental
decomposition theorem
[N, Satz IV] shows that each open ideal in  $R$  is a finite intersection of
parameter ideals (but of course the  $x$'s may vary).  One consequence of
our main result, (4.1), is that a similar statement holds for open monomial
ideals in a Cohen-Macaulay local ring.

Monomial ideals are important in several areas of current research, and they
have been studied in their own right in several papers (for example, \cite{EH}
and \cite{T}), so many useful results are known about such ideals.  In the
present paper we are interested in giving an explicit decomposition of  $I$  as
an irredundant finite intersection of parameter ideals.  We do this in
Section 2 for the special case when  $I=X^n$  ($n$  a positive integer), and it
is shown that  $X^n$  is the irredundant intersection of
the  $\left(\smallmatrix n+d-2\\  d-1\endsmallmatrix\right)$  parameter
ideals  $(x_1^{a_1},\dots, x_d^{a_d})R$, where  $a_1,\dots a_d$  are
positive integers that sum to  $n+d-1$.

To extend this result to an arbitrary monomial ideal  $I$  (such
that  $Rad(I)$ $=$ $Rad(X)$), in Section 3 we introduce and study the  $J$
-corner-elements of a
monomial ideal  $J$.  We show that they are the monomials in  $(J:X)-J$, that
there are only finitely many of them, and that if  $(x_1,\dots,x_{d-1})R$
$\subseteq$ $Rad(J)$, then their  $J$ -residue classes are a minimal
basis, in any order, of  $(J:X)/J$.  Also, if  $Q$  is an open monomial
ideal in a regular local ring  $(R,M)$  of altitude two,
then  $v((Q:X)/Q)$ $=$ $v(Q)-1$  (where  $v(J)$  denotes the
number of elements in a minimal basis of the ideal  $J$), and if  $t$  is an
integer such that  $v(Q)-1$ $\le$ $t$ $\le$ $2v(Q)-1$, then  $Q$  can be
chosen such that  $v(Q:X)=t$.  Finally, we give a geometric interpretation
of  $I$ -corner-elements, an algebraic construction of them, and then close this
section with several examples of such elements.

In Section 4 we show that if  $z_1,\dots z_m$  are the  $I$ -corner-elements,
then  $I$  is the irredundant intersection of the  $m$  parameter
ideals  $P(z_j)$ $=$ $(x_1^{a_{j,1}},\dots x_d^{a_{j,d}})R$,
where  $z_j$ $=$ $x_1^{a_{j,1}-1}\cdots x_d^{a_{j,d}-1}$.
Three interesting corollaries are:  $\cup\{Ass(R/I^n)$;
$n\ge$ $1\}$ $\subseteq$ $Ass(R/X)$;  and, if  $R$  is
a Gorenstein local ring with maximal ideal  $M$, if  $X$  is generated by
a system of parameters, and if  $I$  is open, then $v((I:M)/I)$\break
$=$ $v((I:X)/I)$, and  $I$  is
irreducible if and only if there exists exactly one  $I$\break
-corner-element, and
then  $I$  is generated by a system of parameters.  Also, unique factorization
holds in the sense that if  $I$ $=$ $\cap\{P(z_j)$; $j$
$=$ $1,\dots,m\}$ $=$ $\cap\{P(w_i)$; $i$ $=$ $1,\dots,n\}$, then  $n$
$=$ $m$  and  $\{z_1,\dots z_m\}$ $=$ $\{w_1,\dots,w_n\}$.  Further,
if  $k$  is a positive integer and  $I^{[k]}$  is the ideal generated by
the  $k$-th powers of the monomial generators of  $I$, then  $I^{[k]}$
$=$ $\cap\{(P(z_j))^{[k]}$; $j$ $=$ $1,\dots,m\}$  and
the  $I^{[k]}$ -corner-elements are
the  $m$  monomials  $z_j^{(k)}$ $=$
$x_1^{ka_{j,1}+k-1}\cdots x_d^{ka_{j,d}+k-1}$.

In Section 5 a related decomposition of  $I$  as an irredundant finite
intersection of irreducible ideals is proved.  Specifically, with the notation
of the preceding paragraph, if  $R$  is local with maximal ideal  $M$  and
if  $Q_j$  is maximal in  $S_j$ $=$ $\{Q$; $Q$  is an ideal in  $R$,
$P(z_j)$ $\subseteq$ $Q$, and  $z_j$ $\notin$ $Q\}$  for  $j$
$=$ $1,\dots,m$, then each  $Q_j$  is
irreducible, $\cap\{Q_j$; $j$ $=$ $1,\dots,m\}$  is an irredundant
intersection, and  $(\cap\{Q_j$; $j$ $=$ $1,\dots,m\})$ $\cap$
$(I:X)$ $=$ $I$ $+$ $M(I:X)$.  It then follows that if  $R$  is a regular
local ring and  $X$ $=$ $M$, then  $Q_j=P(z_j)$  for  $j$ $=$ $1,\dots m$.

Finally, in Section 6 we show that: if  $I$  is irreducible, then  $I$  is a
parameter ideal; $I$  is a parameter ideal if and only if  $I$  has exactly one
corner-element;  and, if  $R$  is a Gorenstein local ring of altitude  $d$,
then  $I$  is a parameter ideal if and only if  $I$  is irreducible.  Also, the
parameter ideals that are minimal with respect to containing  $I$  are
the ideals  $P(z)$, where  $z$  is an  $I$ -corner-element.

The authors have been fascinated by the historic and fundamental decomposition
theorems of Emmy Noether, and this fascination gave rise to the results in
[HRS1, HRS2, HRS3, HRS4] and the present paper.  We are pursuing further topics
in this area (in particular, in [HMRS]), and we hope this theory turns out to be
fascinating and useful to others.
\bigskip
\noindent
{\smc{\bf 2. Parametric Decompositions of Powers of an  $R$ -Sequence.}}
The main result in this section, (2.4), shows that if  $X$  is an ideal
generated by an  $R$ -sequence, then  $X^n$  is the irredundant intersection
of  $\left(\smallmatrix n+d-2\\  d-1\endsmallmatrix\right)$  parameter ideals.
To prove this, we need a few preliminary results, so we begin with these.
\medskip
\noindent
{\bf{(2.1) Definition.}}
Let  $R$  be a ring, let  $x_1,\dots,x_d$ $(d>1)$  be an  $R$ -sequence,
and let  $X=(x_1,\dots,x_d)R$.  Then:

{\pmb{(2.1.1)}}  A {\bf monomial} (in  $x_1,\dots x_d$)  is a power
product  $x_1^{e_1}\cdots x_d^{e_d}$, where  $e_1,\dots,e_d$  are nonnegative
integers (so a monomial is either a nonunit or the element 1), and
a {\bf monomial ideal} is a proper ideal generated by monomials.

{\pmb{(2.1.2)}}  A {\bf parameter ideal} (in  $x_1,\dots,x_d$)  is an ideal of the
form  $(x_1^{a_1},\dots, x_d^{a_d})R$, where  $a_1,\dots, a_d$  are positive
integers (so the parameter ideal  $(x_1^{a_1},\dots, x_d^{a_d})R$  is
a monomial ideal generated by the  $R$ -sequence  $x_1^{a_1},\dots, x_d^{a_d})$.
If  $f=x_1^{e_1}\cdots x_d^{e_d}$  is a monomial, then we
let  $\bold{P(f)}$  denote the parameter
ideal  $(x_1^{e_1+1},\dots,x_d^{e_d+1})R$.  (Note that
if  $f=1$, then  $P(f)=X$.)  And if  $a_1,\dots a_d$  are positive integers,
then we let  ${\bold {P(a_1,\dots,a_d)}}$  denote the parameter
ideal  $(x_1^{a_1},\dots, x_d^{a_d})R$  (so  $P(a_1,\dots,a_d)$ $=$ $P(f)$,
where  $f=x_1^{a_1-1}\cdots x_d^{a_d-1})$.
\medskip
\noindent
{\bf{(2.2) Remark.}}
Let  $f$  and  $g$  be monomials.  Then:

{\pmb{(2.2.1)}}  If  $f_1,\dots,f_n$  are monomials
then  $f\in (f_1,\dots,f_n)R$  if and only if  $f\in f_iR$  for some
$i$ $=$ $1,\dots,n$.

{\pmb{(2.2.2)}}  If  $f$ $\in$ $gR$, then there exists a
monomial  $k$  (possibly $k=1$)  such that  $f$ $=$ $gk$.

{\pmb{(2.2.3)}}  If  $h$  is a monomial such that  $fh=gh$, then  $f$ $=$ $g$.

{\pmb{(2.2.4)}} If  $fx_j$ $=$ $gx_i$  for some  $i$ $\not=$
$j$  in  $\{1,\dots,d\}$, then  $f$ $\in$ $x_iR$  and  $g$ $\in$ $x_jR$.

\demo{Proof}  It is shown in [T, Theorem 1] that if  $r$ $\in$ $R$  and  $rf$
$\in$ $(f_1,\dots,f_n)R$, then either  $f$ $\in$ $f_iR$  for some  $i$
$=$ $1,\dots,n$  or  $r$ $\in$ $(x_1,\dots,x_d)R$.  (2.2.1) readily follows
from this.

(2.2.2)--(2.2.4) readily follow by the ``independence'' of power products in
an  $R$ -sequence (that is,  $x_1^{e_1}\cdots x_d^{e_d}$
$=$ $x_1^{a_1}\cdots x_d^{a_d}$  if and only if  $a_i$ $=$ $e_i$  for $i$
$=$ $1,\dots,d$), \qed
\enddemo
\medskip
\proclaim{(2.3) Lemma}
Let $f$ and $g$ be monomials.  Then $g\in P(f)$ (see (2.1.2)) if and only if
$f\notin gR$.
\endproclaim

\demo{Proof}
Let $f=x_1^{e_1}\cdots x_d^{e_d}$.  Then $f\notin x_i^{e_i+1}R$ for $i=1,\dots,d$, since $e_i < e_i+1$ for $i=1,\dots, d$, so (2.2.1) shows that
$f\notin P(f)$.  Therefore if $g\in P(f)$, then $f\notin gR$.

For the converse assume that $g\notin P(f)$ and let $g=x_1^{a_1}\cdots x_d^{a_d}$.  Then $a_i< e_i+1$ for $i=1,\dots, d$, so $e_i\ge a_i$ for
$i=1,\dots,d$, hence $f\in gR$, \qed
\enddemo
\medskip
(2.4), the main result in this section, extends [HRS2, (3.5)] (where it is
shown that in a regular local ring $(R,M=(x,y)R)$,
$M^n$ $=$ $\cap \{(x^{n+1-i},y^i)R$; $i=1,\dots,n\}$).

Concerning the ideals $P(a_1,\dots,a_d)$ in (2.4), see (2.1.2).
\medskip
\proclaim{(2.4) Theorem}
Let $X$ be an ideal that is generated by an $R$ -sequence $x_1,\dots,x_d$
$(d>1)$ and let $n$ be a positive integer.  Then $X^n=\cap\{P(a_1,\dots,a_d);
a_1+\cdots+ a_d= n+d-1\}$ and this intersection is irredundant.  Therefore
$X^n$ is the irredundant intersection of $\left(\smallmatrix n+d-2\\  d-1\endsmallmatrix\right)$ parameter ideals.
\endproclaim

\demo{Proof}
If $n=1$, then this is clear, so it will be assumed that $n>1$.

Let $J=\cap\{P(a_1,\dots,a_d); a_1+\cdots + a_d=n+d-1\}$.  Then since $X^n$
is generated by the monomials $x_1^{e_1}\cdots x_d^{e_d}$, where $e_1,\dots,
e_d$ are nonnegative integers that sum to $n$, to show that $X^n\subseteq J$
it suffices to show that each such monomial is in $J$.  For this, fix
$f=x_1^{e_1}\cdots x_d^{e_d}$  and consider any of the ideals $P(a_1,\dots,a_d)$.
Then $e_i\ge a_i$ for some $i=1,\dots,d$ (since otherwise $n=e_1+\cdots +e_d<n+d=
(e_1+1)+\cdots+(e_d+1)\le a_1+\cdots+a_d=n+d-1$, and this is a contradiction),
so $f\in P(a_1,\dots,a_d)$.  Therefore $f\in J$, so it follows that $X^n\subseteq J$.

For the opposite inclusion, since $Rad(P(a_1,\dots,a_d))$ $=$ $Rad(X)$ for all
positive integers $a_1,\dots,a_d$ that sum to $n+d-1$, [T, Lemma 6] shows
that $J$ is generated by monomials, so it suffices to show that if $f$ is a
monomial in $X-X^n$, then $f$ is not in $J$.  For this let $f=x_1^{e_1}\cdots
x_d^{e_d}\in X^k-X^{k+1}$, where $e_1+\cdots +e_d=k$ and $1\le k<n$.  Then since
$k<n$ there exists a nonnegative integer $h$ such that $(e_1+1)+\cdots+
(e_{d-1}+1)+(e_d+1+h)=n+d-1$,
and since $x_1,\dots,x_d$ is an $R$ -sequence, it follows from
(2.2.1) that $f\notin P((e_1+1),\dots,(e_{d-1}+1),(e_d+1+h))$.  Therefore it
follows that $J\subseteq X^n$, so $J=X^n$.

Also, this intersection is irredundant, since if $\{a_1,\dots,a_d\}$ and
$\{b_1,\dots,b_d\}$ are distinct sets of positive integers that sum to
$n+d-1$, then $b_i > a_i$ for some $i=1,\dots,d$, so
$x_1^{b_1 -1}\cdots x_d^{b_d -1}\in P(a_1,\dots,a_d)$, hence it follows that
$x_1^{b_1 -1}\cdots x_d^{b_d -1}\in \cap\{P(a_1,\dots,a_d)$; $a_1+\cdots + a_d$
$=$ $n+d-1$
and $a_i\not= b_i$ for some $i\}$, and $x_1^{b_1 -1}\cdots x_d^{b_d -1}\notin
P(b_1,\dots,b_d)$, by (2.3), so $x_1^{b_1 -1}\cdots x_d^{b_d -1}\notin \cap
\{P(a_1,\dots,a_d); a_1+\cdots + a_d=n+d-1\}$.

For the final statement, each ideal $P(a_1,\dots,a_d)$ is a parameter ideal,
by (2.1.2).  And the preceding paragraph shows that they are distinct for
distinct $d$-tuples $(a_1,\dots,a_d)$ of positive integers.  To compute
the number of these ideals, since we are only interested in the number of ideals, it may be assumed that $X=(x_1,\dots,x_d)R$ is the maximal ideal $M$
in a regular local ring $(R,M)$.  Then by [HRS2, (2.3.2) and (2.4)] the
number of ideals is $d(X^n)=\dim_{R/M}((X^n:X)/X^n)=\dim_{R/M}(X^{n-1}/X^n)=
v(X^{n-1})=\left(\smallmatrix n+d-2\\  d-1\endsmallmatrix\right)$, \qed
\enddemo
\medskip
\proclaim{(2.5) Corollary}
If $R$ is a Gorenstein local ring and altitude $(R)=d$, then $X^n=\cap\ \{P
(a_1,\dots, a_d)$; $a_1+\cdots + a_d = n+d-1\}$ is an irredundant intersection of
$\left(\smallmatrix n+d-2\\  d-1\endsmallmatrix\right)$ irreducible ideals.
\endproclaim

\demo{Proof}
If $R$ is Gorenstein, then each open parameter ideal is irreducible, so the
conclusion follows from (2.4), \qed
\enddemo
\medskip
\noindent
{\bf{(2.6) Remark.}}
It follows from (2.4) that the cardinality of $\{x_1^{e_1}\cdots x_d^{e_d}$;
$e_1,\dots, e_d$  are positive integers that sum to  $n+d-1\}$  is
$\left(\smallmatrix n+d-2\\  d-1\endsmallmatrix\right)$.
\bigskip
\noindent
{\smc{\bf 3. J-Corner-Elements}}.  We now want to extend (2.4) to an arbitrary
monomial ideal $I$ such that $Rad(I)$ $=$ $Rad(X)$.  (It should be noted that
$Rad(I)$ $=$ $Rad(X)$ is a necessary condition to extend (2.4), since the radical of
each parameter ideal is the radical of $X$, and in (4.1) we show that this
condition is also sufficient.)  To accomplish this extension, we have found it
useful to use ``corner-elements''.  So in this section we introduce such elements and derive some of their basic properties, and then use some of these
properties in the proof of (4.1) to give the desired extension of (2.4).

We think ``corner-elements will be of interest and use in other problems, so
in this section we prove several results
concerning them.  Specifically, we show in (3.2) and (3.7) that if $J$ is a
monomial ideal, then there exist only finitely many $J$ -corner-elements,
that they are the monomials in $(J:X)-J$, and that if  $(x_1,\dots, x_{d-1})R$
$\subseteq$ $Rad(J)$, then the $J$ -residue classes of these corner-elements are a
minimal basis, in any order, of $(J:X)/J$.  We then apply these results to the case
when $J$ is an open monomial ideal in a regular local ring $(R,M)$ of
altitude two, give a geometric interpretation of $I$ -corner-elements and an
algebraic construction of them, and then close this section with several examples of such elements.

We begin with the definition.
\medskip
\noindent
{\bf{(3.1) Definition.}}
Let $J$ be a monomial ideal.  Then a $J$ -{\bf corner-element} is a monomial
$z$ such that $z\notin J$ and $zx_i\in J$ for $i=1,\dots,d$.
\medskip
(The name ``corner-element'' is suggested by the geometric interpretation
in (3.13), where a corner-element is an element $z=x^ay^b$ with coordinates
$(a,b)$ such that $(a,b+1),(a+1,b)$, and $(a+1,b+1)$ are the coordinates of
points in $I$ and $z\notin I$.)

Concerning (3.1), note that $1$ is the unique $X$ -corner-element (since each
nonunit monomial is in $X$).  Also, if $J$ is a monomial ideal and $1$ is a
$J$ -corner-element, then $1x_i\in J$ for $i=1,\dots, d$, so $J=X$.

In (3.2) we characterize the $J$ -corner-elements and show that there are only
finitely many of them.  (It follows from (3.2) that $J$ uniquely determines its
corner-elements.  The converse of this is proved in (4.2) when $Rad(J)$
$=$ $Rad(X)$.)
\medskip
\proclaim{(3.2) Proposition}
If $J$ is a monomial ideal, then the $J$ -corner-elements are the monomials in
$(J:X)-J$.  Also, if $z,z'$ are distinct $J$ -corner-elements, then
$zR\not\subseteq z'R$ and $z'R\not\subseteq zR$, so there exist only finitely
many $J$ -corner-elements.
\endproclaim

\demo{Proof}
Let $\bold C$ be the set of $J$ -corner-elements (so each element in $\bold C$ is a
monomial).  Then it is clear from (3.1) that $\bold C\subseteq (J:X)-J$.  And if
$z$ is a monomial in $(J:X)-J$, then $z\notin J$ and $zx_i\in J$
for $i=1,\dots,d$, so $z$ is a $J$ -corner-element, hence
$z\in \bold C$.  Therefore $\bold C$ is the set of monomials in $(J:X)-J$.

Now let $z$ and $z'$ be distinct $J$ -corner-elements and suppose that
$zR\subseteq z'R$.  Then (2.2.2) shows that $z=z'f$ for some monomial $f$
(and $f\not= 1$, since $z\not=z'$).  But this implies that $z=z'f\in J$
(since $z'$ is a $J$ -corner-element), and this is a contradiction.  Therefore
$zR\not\subseteq z'R$ and, similarly, $z'R\not\subseteq zR$.

Finally, the ideal generated by the $J$ -corner-elements (viewed as elements
in $Z_k[x_1,\dots,x_d]$, where $k$ is the characteristic of $R$ and where
$Z_k$ is the ring generated by the identity of $R$) is finitely generated, so
since there are no inclusion relations among the ideals they generate, (2.2.1)
shows that there are only finitely many of them, \qed
\enddemo
\medskip
\proclaim{(3.3) Corollary}
Let $J$ be a monomial ideal and let $z_1,\dots,z_m$ be the $J$ -corner-elements.
Then for $j=1,\dots,m$ it holds that $(z_1,\dots,z_{j-1},z_{j+1},\dots,z_m)R
\subseteq P(z_j)$ and $z_j\notin P(z_j)$.  Therefore $\cap\{P(z_j); j=1,\dots,
m\}$ is an irredundant intersection of parameter ideals.
\endproclaim

\demo{Proof}
(It follows from (3.2) that there are only finitely many $J$ -corner-elements.
Also, if $m=1$ and $z_1=1$, then $P(z_1)=X$, $(0)$ (the ideal generated by
the empty set) is contained in $X$, and $1\notin P(1)=X$, so the conclusion
holds in this case.)

Fix $j$ $\in$ $\{1,\dots,m\}$.  Then it follows from (3.2) that
if  $i$ $\in$ $\{1,\dots,j-1,\break
j+1,\dots,m\}$, then $z_j$ $\notin$ $z_iR$, so (2.3) shows
that  $z_i$ $\in$ $P(z_j)$  (hence  $(z_1,\dots,z_{j-1},\break
z_{j+1},\dots,z_m)R$ $\subseteq$ $P(z_j)$)  and  $z_j$
$\notin$ $P(z_j)$.  This shows that $\cap\{P(z_j);j=1,\dots,m\}$ is an
irredundant intersection, and (2.1.2) shows that the ideals $P(z_j)$ are parameter ideals, \qed
\enddemo
\medskip
In (3.4) we specify the $X^n$ -corner-elements.
\medskip
\proclaim{(3.4) Corollary}  If $n>1$ is a positive integer, then the $X^n$ -corner-elements are the $\left(\smallmatrix n+d-2\\  d-1\endsmallmatrix
\right)$ generators $x_1^{e_1}\cdots x_d^{e_d}$ of $X^{n-1}$ (so
$e_1,\dots,e_d$ are nonnegative integers such that $e_1+\cdots+ e_d=n-1$).
\endproclaim

\demo{Proof}
By (3.2) the $X^n$ -corner-elements are the monomials in $X^{n-1}-X^n$ (since
$X^n:X=X^{n-1}$), and since $X$ is generated by an $R$ -sequence of length $d$
it follows that there are $\left(\smallmatrix n+d-2\\  d-1\endsmallmatrix\right)$ distinct such elements, \qed
\enddemo
\medskip
It follows from (3.5) that if  $z$  is a  $J$ -corner-element, then
the  $d$  elements  $zx_1,\dots, zx_d$  are members of distinct
principal ideals generated by
monomials in $J$.
\medskip
\proclaim{(3.5) Proposition}
Let $f$ and $g$ be monomials and let $i\not= j\in\{1,\dots,d\}$.  If
$fx_i\in gR$ and $fx_j\in gR$, then $f\in gR$.
\endproclaim

\demo{Proof}
If $fx_i\in gR$ and $fx_j\in gR$, then (2.2.2) shows that there exist
monomials $h_i$, $h_j$ such that $fx_i=gh_i$ and $fx_j=gh_j$.  Then
$fx_ix_j=gh_ix_j=gh_jx_i$, so $h_ix_j=h_jx_i$ by (2.2.3).  (2.2.4) then shows
that $h_i\in x_iR$, so $h_i=kx_i$ for some monomial $k$  by (2.2.2).  Therefore
$fx_ix_j=gh_ix_j=g(kx_i)x_j$, so $f=gk\in gR$ by (2.2.3), \qed
\enddemo
\medskip
\proclaim{(3.6) Corollary}
If $J$ is a monomial ideal that has a corner-element, and if $f_1,\dots,f_n$
are monomials that generate $J$, then $n\ge d$.
\endproclaim

\demo{Proof}
This follows immediately from (3.5), \qed
\enddemo
\medskip
In (3.7) it is shown that if $(x_1,\dots,x_{d-1})R$ $\subseteq$ $Rad(J)$,
then the
$J$ -residue classes of the $J$ -corner-elements are a minimal basis, in any
order, of $(J:X)/J$.  (In this regard, note that if $J:X=J$, then there are no
$J$ -corner-elements, and the empty set does generate the ideal $(J:X)/J=J/J$.
On the other hand, if $Rad(J)$ $=$ $Rad(X)$ then $J:X\not=J$.)
\medskip
\proclaim{(3.7) Theorem}
Let $J\not=X$ be a monomial ideal such that $(x_1,\dots,x_{d-1})R$
$\subseteq$ $Rad(J)$.  Then the $J$ -residue classes of the
$J$ -corner-elements are a minimal basis, in any order, of $(J:X)/J$.
\endproclaim

\demo{Proof}
(By ``minimal basis'', we mean a basis such that no proper subset is a
generating set of the ideal.)  Since $Rad((x_1,\dots,x_{d-1})R)$
$\subseteq$ $Rad(J)$,
[T, Theorem 6] shows that $J:X$ is a monomial ideal, so it follows from (3.2)
that the $J$ -residue classes of the $J$ -corner-elements generate $(J:X)/J$.

Let $\bold C=\{z_1,\dots, z_m\}$ be the set of $J$ -corner-elements ($\bold C$ is a finite
set by (3.2)) and suppose that there exists a permutation $\pi 1,\dots,\pi m$
of $1,\dots,m$ such that $z_{\pi1}\in (z_{\pi2},\dots,z_{\pi m})R$.  Then
$z_{\pi 1}\in z_{\pi k}R$ for some $k=2,\dots,m$ by (2.2.1), so (2.2.2) shows
that $z_{\pi 1}=z_{\pi k}f$ for some monomial $f$ ($f\not= 1$, since
$z_1,\dots,z_m$ are distinct).  However, this implies that
$z_{\pi 1}=z_{\pi k}f\in J$ (since $z_{\pi k}$ is a $J$\break
-corner-element), and 
this is a contradiction.  Therefore $z_{\pi 1}\notin (z_{\pi2},\dots,z_{\pi m})R$
for all permutations $\pi 1,\dots,\pi m$ of $1,\dots,m$.
And no  $z_j$  in in  $J$, so it follows from (2.2.1) that
the  $J$ -residue classes of  $z_1,\dots,z_m$  are a minimal basis,
in any order, of $(J:X)/J$, \qed
\enddemo
\medskip
In (3.8) we consider the case when $J=Q$ is open in a regular local ring  $R$  of
altitude two.  ((3.8) was noted in [HRS2, (3.3)] for the case $M=(x,y)R$,
and therein a homological proof using [HS, (2.1)] was sketched for an arbitrary
open ideal (in an altitude two regular local ring).  (3.8) gives a non-homological proof for an arbitrary $R$ -sequence of length two, but only for
the case of an open monomial ideal.)
\medskip
\proclaim{(3.8) Corollary}
Let $(R,M)$ be a regular local ring of altitude two, let $x,y$ be an
$R$ -sequence, and let $Q\not=(x,y)R$ be an open monomial ideal in $x$ and $y$,
say $v(Q)=n$.  Then $v((Q:X)/Q)=n-1$.
\endproclaim

\demo{Proof}
Let $Q=(f_1,\dots,f_n)R$ and lexicographically order the $f_i$ by saying that
$f_i<f_j$ (for $f_i=x^ay^b$ and $f_j=x^cy^e$) if either $a<c$ or $a=c$ and
$b<e$.  Then it may be assumed that $f_1<f_2<\cdots <f_n$.  Therefore, since
$v(Q)=n$ and $Q$ is open, it follows that there exist positive integers
$h,k,h_2<\cdots< h_{n-1}$ $(h_{n-1} \le h-1)$, and $k_{n-1}<\cdots< k_2$ $(k_2 \le k-1)$
such that $f_1=y^h$, $f_n=x^k$, and $f_i=x^{k-k_i}y^{h-h_i}$ for
$i=2,\dots,n-1$.  For $j=1,\dots,n-1$ let $z_j=x^{k-k_{j+1}-1} y^{h-h_j-1}$
(with $h_1=0=k_n$).  Then $z_j\notin Q$, $z_jx=x^{k-k_{j+1}}y^{h-h_j-1}\in
f_{j+1}R\subseteq Q$, and $z_jy=x^{k-k_{j+1}-1}y^{h-h_j}\in f_jR\subseteq Q$, so
$z_j\in (Q:X)-Q$ for $j=1,\dots, n-1$.  Thus each $z_j$ is a $Q$ -corner-element,
and the geometric interpretation in (3.13) shows that every $Q$ -corner-element
is one of these $z_1,\dots,z_{n-1}$.  Therefore the conclusion follows from
(3.7), \qed
\enddemo
\medskip
For the next corollary of (3.7) we need the following definition.
\medskip
\noindent
{\bf{(3.9) Definition.}}
If $J$ is a monomial ideal, then $\bold {c(J)}$ denotes the number of $J$\break
-corner-elements.
\medskip
\proclaim{(3.10) Corollary}
With the notation of (3.8), there exists a polynomial $p(x)$ of degree two
such that $p(n)=c(Q^n)$ for large $n$.
\endproclaim

\demo{Proof}
It is well known that there exists a polynomial $q(x)$ of degree two such that
$q(n)=v(Q^n)$ for large $n$.  But $Q^n$ is a monomial ideal, so the conclusion
follows immediately from (3.7) and (3.8) with $p(x)=q(x)-1$, \qed
\enddemo
\medskip
In (3.11) it is shown that if  $v(Q)$ $=$ $n$, where  $Q$  is as in (3.8),
then  $n-1$ $\le$ $v(Q:X)$ $\le$ $2n-1$  and for each
integer  $t$  between $n-1$  and  $2n-1$  the ideal  $Q$  can be chosen so
that  $v(Q:X)$ $=$ $t$.
\medskip
\proclaim{(3.11) Proposition}
With the notation of (3.8) assume that $v(Q)=n$.  Then $n-1\le v(Q:X)\le 2n-1$,
and for each intermediate integer $t$ there exists an ideal $Q$ such that
$v(Q)=n$ and $v(Q:X)=t$.
\endproclaim

\demo{Proof}
(3.8) shows that $(Q:X)/Q$ is generated by $v(Q)-1=n-1$ elements, so it follows
that $Q:X$ can be generated by the preimages of these $n-1$ elements together
with the $n$ generators of $Q$.  Therefore $n-1\le v(Q:X)\le 2n-1$.

Now let  $t$  be a given positive
integer such that  $n-1\le t\le 2n-1$  and let  $s$  be the integer such
that $t=(n-1)+s$, so $0\le s\le n$.  For $i=1,\dots,s$ let $f_i=x^{2(i-1)}y^{n+s-2i}$, for $i=s+1,\dots,n$ let $f_i=x^{s+i-1}y^{n-i}$, and
let $Q=(f_1,\dots,f_n)R$.  Then the $Q$ -\CES are the elements $z_j=x^{2j-1}y^{n+s-2j-1}$ (for $j=1,\dots,s)$ and the elements
$z_j=x^{s+j-1}y^{n-1-j}$ (for $j=s+1,\dots,n-1$).  If $s=0$, then $f_1\in z_1R$, $f_i\in z_{i-1}R\cap z_iR$ for $i=2,\dots,n-1$, and $f_n\in z_{n-1}R$, and
if $s>0$, then $f_i\in z_{i-1}R$ for $i=s+1,\dots,n$ and $f_i\notin (z_1,\dots,
z_{n-1})R$ for $i=1,\dots,s$, so by (2.2.1) (and (3.8)) it readily follows that
$f_1,\dots,f_s, z_1,\dots z_{n-1}$ is a minimal basis of $Q:X$ so
$v(Q:X)=s+n-1=t$, \qed
\enddemo
\medskip
\proclaim{(3.12) Corollary}
Let $(R,M=(x,y)R)$ be a regular local ring of altitude two, let $Q\not= M$
be an open monomial ideal in $x$ and $y$, and let $n=v(Q)$.  Then
$v((Q:M)/Q)=n-1$, $n-1\le v(Q:M)\le 2n-1$, and for each intermediate integer
$t$ there exists an ideal $Q$ such that $v(Q)=n$ and $v(Q:M)=t$.
\endproclaim

\demo{Proof}
This follows immediately from (3.8) and (3.11), since $M=X$, \qed
\enddemo
\medskip
In (3.13) we give a geometric interpretation of the $I$ -\CES for a monomial
ideal $I$ in an $R$ -sequence $x,y$ of length two such that $Rad(I)$
$=$ $Rad((x,y)R$).
\medskip
\noindent
{\smc{\bf (3.13) Geometric Interpretation}}.
Assume that $d=2$, let $x=x_1$ and $y=x_2$, let $f_1,\dots,f_n$ be a minimal
basis of $I$ (where the $f_l$ are monomials in $x$ and $y$, say $f_l=x^{i_l}
y^{j_l}$), and assume that $Rad(I)$ $=$ $Rad((x,y)R)$.  Lexicographically order the
$f_l$ (as in the proof of (3.8)) and assume that $f_1<\cdots < f_n$.  Plot the
$n$ points $(i_l,j_l)$ (corresponding to the $f_l$) in the first quadrant of
the $xy$-plane.  Then for each of these $n$ points draw the horizontal line
segment connecting $(i_l,j_l)$, $(i_l+1,j_l)$, $(i_l+2,j_l),\dots,$ and draw
the vertical line segment connecting $(i_l,j_l)$, $(i_l,j_l+1)$, $(i_l,j_l+2),\dots$.  (Then it is clear that there is a one-to-one correspondence from the set $\bold D =\{(a,b); a\ge i_l$ and $b\ge j_l \text{ for some } l=1,\dots,n\}$ to a subset $\bold M$ of the set of monomials in $Q$, and it
follows from (2.2.1) that, in fact, every monomial in $Q$ is in $\bold M$.)  Since
$(i_l,j_l)< (i_{l+1},j_{l+1}),(i_{l+1},j_l)$ are the coordinates of the
intersection of the rightward extending horizontal line segment thru $(i_l,j_l)$ with the ascending vertical line segment thru $(i_{l+1},j_{l+1})$.  Then
$z_l=x^{i_{l+1}-1}y^{j_l-1}\notin Q$, $z_ly$ has coordinates on the
rightward extending horizontal line segment thru $(i_l,j_l)$ (so $z_ly\in Q$),
and $z_lx$ has coordinates on the ascending vertical line segment thru
$(i_{l+1},j_{l+1})$ (so $z_lx\in Q$), hence $z_l$ is a $Q$ -corner-element.
And since a $Q$ -\CE must correspond to some $(a,b)$ with $0\le a < i_n$ and $0\le b<j_1$,
it is readily checked that all $Q$ -\CES are obtained in this way, so there are
exactly $n-1$ of them, where $n=v(Q)$.
\medskip
\noindent
{\smc{\bf (3.14) Algebraic Construction}}.
Let   $x_1,\dots,x_d$  be an  $R$ -sequence and let  $I$  be
a monomial ideal such that  $Rad(I)$ $=$ $Rad((x_1,\dots,x_d)R)$.  Then
the following is an algebraic construction of the  $I$ -corner-elements.
(For ease of
description it will be said that $\deg(f)=n$ if $f=x_1^{e_1}\cdots x_d^{e_d}$
and $e_1+\cdots + e_d=n$.)  Let $S$ be the set of monomials (in $x_1,\dots, x_d)$
that are not in $I$ (so $S$ is a finite set, since for $i=1,\dots,d$ there
exists a positive integer $n_i$ such that $x_i^{n_i}\in I$).  Let
$w=\max\{n;n=\deg(f)\text{ for some } f\in S\}$.  For $j=1,\dots,w$ let
$D_j=\{f\in S; \deg(f)=j\}$, let $C_w=D_w$, and for $j=1,\dots,w-1$ let
$C_j=\{f\in D_j; fx_i\notin D_{j+1} \text{ for } i=1\dots, d\}$ (possibly
some of the sets $C_j$ are empty for $j<w$).  Then $C_l\cup\cdots\cup C_w$ is
the set of $I$ -\CES (and this union is disjoint).

\demo{Proof}
Let  $f\in C_j$  for some  $j=1,\dots,w$.
Then  $f\notin I$  (since  $f\in C_j\subseteq S$)  and for  $i=1,\dots,d$  it
holds that  $\deg(fx_i)=j+1$.  If  $j=w$,
then  $fx_i\notin S$  (for no element in  $S$  has degree greater
than  $w=j$), and if  $j<w$,
then  $fx_i\notin D_{j+1}=\{g\in S; \deg(g)=j+1\}$  (by the definition
of  $C_j$).  Therefore in either
case  $(j=w \text{ or } j<w)fx_i\notin S$  for  $i=1,\dots,d$,
so  $fx_i\in I$, hence  $f$  is an  $I$ -corner-element.
Therefore  $C_1\cup\cdots\cup C_w$ $\subseteq$ $\bold C$ $=$ $\{f; f$  is
an  $I$ -corner-element\}.

And if $g\in \bold C$, then $g\notin I$, so $g\in S$, so $g\in D_j$,
where $j=\deg(g)$.
Also, $\deg(gx_i)=j+1$ and $gx_i\in I$ for $i=1,\dots,d$, so $gx_i\notin D_{j+1}$.
Therefore $g\in C_j$, so $\bold C\subseteq C_1\cup\cdots\cup C_w$, \qed
\enddemo
\medskip
\noindent
{\bf{(3.15) Remark.}}
With the notation of (3.14) let  $f$  be a monomial that is not in  $I$.  Then
there exists a monomial  $g$  (possibly  $g=1$) such that  $fg$  is
an  $I$ -corner-element.

\demo{Proof}
It may be assumed that $f$ is not an $I$ -corner-element,
so $fx_i\notin I$ for some
$i=1,\dots,d$.  Let $T=\{g; g \text{ is a monomial in } x_1,\dots,x_d
\text{ and } fg\notin I\}$.  Then $T$ is a finite set (since $T$ is contained
in the finite set $S$ of (3.14)), so let $g\in T$ such that the sum of its
exponents is greater than or equal to the sum of the exponents of the other
monomials in $T$.  Then $fgx_i\in I$ for $i=1,\dots,d$, by the maximality of
the sum of the exponents of $g$, so $fg$ is an $I$ -corner-element, \qed
\enddemo
\medskip
Before giving some examples of $I$ -corner-elements,
we first prove one more result
concerning them.  (Some additional properties are given in (4.11)-(4.12).)
\medskip
\proclaim{(3.16) Proposition}
Let $I\subset J$ be monomial ideals such that $Rad(I)$ $=$ $Rad(X)$.  Then some 
$I$ -\CE is in $J$.
\endproclaim

\demo{Proof}
There exists a monomial $f$ $\in$ $J-I$, by hypothesis.  Then (3.15) shows
that there exists a monomial $g$ (possibly $g=1$) such that $fg$ is an
$I$ -corner-element, and it is clear that $fg\in J$, \qed
\enddemo
\medskip
This section will be closed with several examples of $Q$ -\CES for an open
monomial ideal $Q$ in a regular local ring.
\medskip
\noindent
{\bf{(3.17) Example.}}
Let $(R,M=(x,y)R)$ be a regular local ring of altitude two, let
$x_1=x$ and $x_2=y$, and let $Q=(y^9,xy^7,x^3y^4,x^5y^2,x^{11})R$, so $f_1=y^9$,
$f_2=xy^7$, $f_3=x^3y^4$, $f_4=x^5y^2$, $f_5=x^{11}$.  Then the
$Q$ -\CES are $z_1=y^8$, $z_2=x^2y^6$, $z_3=x^4y^3$, and $z_4=x^{10}y$.  (This
can be checked by using either (3.13) or (3.14).)  Therefore
$(Q:M)/Q=(y^8,x^2y^6,x^4y^3,x^{10}y)R/Q$ by (3.7).
\medskip
\noindent
{\bf{(3.18) Example.}}
Let $(R,M=(x,y,z)R)$ be a regular local ring of altitude three, let
$x_1=x$, $x_2=y$, and $x_3=z$, and let $Q=(z^4,y^2z^3,y^3,xyz,xy^2,x^2)R$.
Then the $Q$ -\CES are $yz^3$, $y^2z^2$, $xz^3$, and $xy$.  (This can be checked
by writing down the sets $S$, $D_j$, and $C_j$ (for $j=1,\dots,4)$ of (3.14).
Thus  $S$ $=$ $\{z,z^2,z^3,y,yz,\break
yz^2,yz^3,y^2,y^2z,y^2z^2,x,xz,xz^2,xz^3,xy\}$  (in
lexicographic order), $D_1$ $=$ $\{z,y,x\}$,\linebreak
$D_2$ $=$ $\{z^2,yz,y^2,xz,xy\}$, 
$D_3$ $=$ $\{z^3,yz^2,y^2z,xz^2\}$, and $D_4$ $=$ $\{yz^3,y^2z^2,xz^3\}$.  Then
$C_4$ $=$ $D_4$, $C_3$ $=$ $\emptyset$ (since at least one of $fx$, $fy$, $fz$ is in $D_4$
for each $f\in D_3$), $C_2$ $=$ $\{xy\}$ (since at least one of $fx$, $fy$, $fz$
is in $D_3$ for $f\in \{z^2,yz,y^2,xz\}$ and none of $xyx, xy^2, xyz$ is in
$D_3$) and $C_1$ $=$ $\emptyset$ (since at least one of $fx, fy, fz$ is in $D_2$ for
each $f\in D_1$).)
\medskip
\noindent
{\bf{(3.19) Example.}}
Let  $(R,M=(w,x,y,z)R)$  be a regular local ring of altitude four,
let  $x_1=w$, $x_2=x$, $x_3=y$, and  $x_4=z$, and
let  $Q$ $=$ $(z^5,yz^4,y^2z^2,y^3,xz^2,xyz,$\linebreak
$x^3z,x^3y^2,x^4,w)R$.
Then the  $Q$ -\CES are  $z^4, yz^3, y^2z,x^2z, x^2y^2$, and  $x^3y$.
(This can be checked by using (3.14).)
\medskip
\noindent
{\bf{(3.20) Example.}}
Let $(R,M=(w,x,y,z)R)$ be a regular local ring of altitude four, let
$x_1=w$, $x_2=x$, $x_3=y$, and $x_4=z$, and let
$Q=(x^d,y^c,x^b,wxyz, w^a)R$, where $a>1$, $b>1$, $c>1$, $d>1$ are integers.
Then the $Q$ -\CES are $x^{b-1}y^{c-1}z^{d-1}$, $w^{a-1} y^{c-1} z^{d-1}$,
$w^{a-1}x^{b-1}z^{d-1}$, and $w^{a-1}x^{b-1}y^{c-1}$.  (This can be checked
by using (3.14).)
\medskip
\noindent
{\bf{(3.21) Example.}}
Let $(R,M=(x_1,\dots,x_d)R)$ be a regular local ring of altitude $d$ and let
$Q=(x_1^{a_1},\dots, x_d^{a_d})R$, where the $a_i$ are positive integers.  Then
$Q$ is irreducible, so by (4.3) there is only one $Q$ -corner-element,
namely $z=x_1^{a_1-1}
\cdots x_d^{a_d-1}$.
\bigskip
\noindent
{\smc{\bf 4. Parametric Decompositions of Monomial Ideals.}}
(2.4) (together with (3.4)) shows that $X^n$ is the irredundant finite
intersection of the parameter ideals $P(z)$,
where  $z$  is an  $X^n$ -corner-element.  The
main result in this section, (4.1), generalizes this to an arbitrary monomial
ideal $I$ such that $Rad(I)$ $=$ $Rad(X)$.  And in (4.10) we show that such a
decomposition is unique.
\medskip
\proclaim{(4.1) Theorem}
Let $I$ be a monomial ideal such that $Rad(I)$ $=$ $Rad(X)$ and let
$z_1,\dots,z_m$ be the $I$ -corner-elements.
Then $I=\cap\{P(z_j); j=1,\dots,m\}$ is a
decomposition of $I$ as an irredundant intersection of parameter ideals.
\endproclaim

\demo{Proof}
Let $J=\cap \{P(z_j); j=1,\dots,m\}$.  Then (3.3) shows that $J$ is the
irredundant intersection of the $m$ parameter ideals $P(z_j)$.

Now let $f$ be a monomial in $I$ and suppose that $f\notin P(z_j)$ for
some $j=1,\dots,m$.  Then $z_j\in fR\subseteq I$, by (2.3), and this contradicts
the fact that $z_j\notin I$ (since $z_j$ is an $I$ -corner-element).
Therefore $I\subseteq
J$.

Finally, [T, Lemma 6] shows that $J$ is a monomial ideal (since $Rad(P(z_j))$
$=$ $Rad(X)$ for $j=1,\dots,m)$, so it suffices to show that each
monomial that is not in $I$ is not in $J$.  For this, let $f$ be a monomial
that is not in $I$.  Then (3.15) shows that there exists a monomial $g$
(possibly $g=1$) such that $fg$ is an $I$ -corner-element,
so $fg=z_j$ for some
$j=1,\dots,m$ (since (3.2) shows that the $I$ -\CES are finite in number and
uniquely determined by $I$).  Then $f\notin P(z_j)$, by (2.3), so it follows
that $I\supseteq J$, hence $I=J$ by the preceding paragraph, \qed
\enddemo
\medskip
In (4.2) it is shown that the \CES of a monomial ideal $I$ determine $I$ when
$Rad(I)$ $=$ $Rad(X)$.
\medskip
\proclaim{(4.2) Corollary}
If $I$ and $J$ are monomial ideals such that $Rad(I)$ $=$ $Rad(X)$
$=$ $Rad(J)$ and if
$(I:X)-I=(J:X)-J$, then $I=J$.
\endproclaim

\demo{Proof}
If $(I:X)-I=(J:X)-J$, then $I$ and $J$ have the same corner-elements,
by (3.2), so this
follows immediately from (4.1), \qed
\enddemo
\medskip
\proclaim{(4.3) Corollary}
If $Q$ is an open monomial ideal in a Gorenstein local ring  $R$  of altitude $d>1$,
then $Q$ is irreducible if and only if there exists exactly
one  $Q$ -corner-element, and
then $Q$ is generated by a system of parameters.
\endproclaim

\demo{Proof}
Let $m$ be the number of $Q$ -corner-elements.
Then $Q$ is the irredundant intersection of
$m$ (open) parameter ideals, by (4.1).  Since $R$ is Gorenstein, an open
parameter ideal is irreducible, so $Q$ is the irredundant intersection of $m$
(open) irreducible ideals.  Since each such
decomposition of $Q$ has the same number of factors, $m=1$ if and only if
$Q$ is irreducible.

For the final statement, if $Q$ is irreducible, then $Q=P(z)$ is generated
by a system of parameters, where $z$ is the $Q$ -corner-element, \qed
\enddemo
\medskip
The next corollary is closely related to (2.5) and (4.3).
\medskip
\proclaim{(4.4) Corollary}
Let $I$ and $z_1,\dots,z_m$ be as in (4.1) and assume that $R$ is a
Gorenstein local ring of altitude $d$.  Then $I=\cap\{P(z_j); j=1,\dots,m\}$
is an irredundant intersection of $m$ irreducible ideals.
\endproclaim

\demo{Proof}
If $R$ is Gorenstein, then each open parameter ideal is irreducible, so the
conclusion follows from (4.1), \qed
\enddemo
\medskip
In [T, Theorem 8] it is shown (among other things) that if $Rad(I)$
$=$ $Rad(X)$,
then $\cup\{P; P \in Ass(R/I)\}=\cup\{Q;Q\in Ass(R/X)\}$.
(4.1) yields a simple proof of the following closely related result.
\medskip
\proclaim{(4.5) Corollary}
If $I$ is as in (4.1), then $\cup\{Ass(R/I^n); n\ge 1\}\subseteq
Ass(R/X)$.
\endproclaim

\demo{Proof}
It is well known that if $Y$ and $Z$ are ideals that are generated by $R$ -sequences such that $Rad(Y)$ $=$ $Rad(Z)$, then $Ass(R/Y)=Ass(R/Z)$.
It therefore follows that if $z_1,\dots,z_m$ are the $I$ -corner-elements, then
$Ass(R/X)=Ass(R/P(z_j))$ for $j=1,\dots,m$ (since each
$P(z_j)$ is generated by powers of $x_1,\dots,x_d$).  Therefore, since
$I=\cap\{P(z_j);j=1,\dots,m\}$, it follows that $Ass(R/I)=
Ass(R/(\cap\{P(z_j); j=1,\dots,m\}))\subseteq \cup\{Ass(R/P(z_j)); j=1,\dots,m\}=Ass(R/X)$.  Finally, $I^n$ is generated by monomials for
all $n\ge 1$, so it follows from what was just shown that
$Ass(R/I^n)\subseteq Ass(R/X)$, \qed
\enddemo
\medskip
For the proof of the next corollary we need the following definition.
\medskip
\noindent
{\bf{(4.6) Definition.}}
If $P$ is a prime divisor of an ideal $I$ in a Noetherian ring, then
$\bold{D_P(I)}$ denotes the number of $P$ -primary ideals in a decomposition of
$I$ as an irredundant intersection of irreducible ideals.
\medskip
Concerning (4.6), a classical result of E. Noether [N, Satz VII] says that
$D_P(I)$ is well defined (that is, $D_P (I)$  is independent of the particular irredundant irreducible decomposition
of $I$).
\medskip
\proclaim{(4.7) Corollary}
Let $(R,M)$ be a Gorenstein local ring, let $X$ be an ideal generated by a
system of parameters $x_1,\dots,x_d$, and let $Q$ be an open monomial ideal.
Then $v((Q:M)/Q)=v((Q:X)/Q)$.
\endproclaim

\demo{Proof}
Let $m$ be the number of $Q$ -corner-elements.  Then (3.7) shows that\linebreak
$v((Q:X)/Q)$ $=$ $m$, and (4.1) shows that $Q$ is the
irredundant intersection of
$m$ parameter ideals.  Since $R$ is Gorenstein, each of these parameter ideals
is irreducible, so $Q$ is the irredundant intersection of $m$ irreducible
ideals, so $D_M(Q)=m$ (see (4.6)).  However, [HRS2, (2.4)] shows that
$D_M(Q)=v((Q:M)/Q)$.  Therefore $v((Q:X)/Q)=m=v((Q:M)/Q)$, \qed
\enddemo
\medskip
(4.1) shows that the $I$ -\CES determine a decomposition of $I$ as an
irredundant intersection of parameter ideals.  (4.8) shows that the converse
also holds.
\medskip
\proclaim{(4.8) Proposition}
For $j=1,\dots,m$ let $\bold a_j=(a_{j,1},\dots,a_{j,d})$ be a $d$-tuple
of positive integers and let $I=\cap\{P(\bold a_j); j=1,\dots,m\}$ be a
decomposition of $I$ as an irredundant intersection of parameter ideals.
Then the $I$ -\CES are the $m$ elements $x_1^{a_{j,1}-1}\cdots x_d^{a_{j,d}-1}$.
\endproclaim

\demo{Proof}
(Note:  $Rad(I)$ $=$ $Rad(X)$, since $Rad(P(\bold a_j))$
$=$ $Rad(X)$ for $j$ $=$ $1,\dots,m$.)

It will first be shown that each of the $m$ elements $z_j=x_1^{a_{j,1}-1}\cdots
x_d^{a_{j,d}-1}$ is an $I$ -corner-element.

For this, note first that $P(z_j)=P(\bold a_j)$ for $j=1,\dots,m$.  Therefore
$z_i\notin z_jR$ for all $i\not=j\in\{1,\dots,m\}$ (for if
$z_i\in z_jR$, then $P(\bold a_i)=P(z_i)\subseteq P(z_j)=P(\bold a_j)$, and
this is a contradiction).  Therefore (2.3) shows that $z_j\notin P(z_j)=P(\bold a_j)$ (so $z_j\notin I$) and that $z_j\in P(z_k)=P(\bold a_k)$ for
$k\in \{1,\dots,j-1,j+1,\dots,m\}$.  Also, $z_jx_i\in x_i^{(a_{j.1}-1)+1}R
\subseteq P(\bold a_j)$ for $i=1,\dots,d$, so $z_jx_i\in\cap\{P(\bold a_h); h=1,\dots,m\}=I$.  Therefore $z_j$ is an $I$ -corner-element, so it follows that
$z_1,\dots,z_m$ are among the $I$ -corner-elements.

Now let $w$ be an $I$ -corner-element.  Then $w\notin I$, so $w\notin P(z_j)=P(\bold a_j)$ for
some $j=1,\dots,m$.  Therefore $z_j\in wR$, by (2.3), so $z_j=wg$ for
some monomial $g$ by (2.2.2).  If $g\not=1$, then $wg\in I$, since $w$ is an
$I$ -corner-element.  But this implies that $z_j\in I$, and this contradicts the fact that
$z_j$ is an $I$ -corner-element.  Therefore $g=1$, so $w=z_j$, so $z_1,\dots,z_m$ are all
the $I$ -corner-elements, \qed
\enddemo
\medskip
\proclaim{(4.9) Corollary}
Let $z_1,\dots,z_m$ be monomials such that $z_i\notin z_jR$ for $i\not=
j\in\{1,\dots, m\}$, let $J=(z_1,\dots,z_m)R$, and let $I=\cap\{P(z_j); j=1,\dots,m\}$.  Then $z_1,\dots,z_m$ are the $I$ -\CES and $I:X=I+J$.
\endproclaim

\demo{Proof}
If $\cap \{P(z_j); j=1,\dots,m\}$ is an irredundant intersection, then it
follows from (4.8) that the $I$ -\CES are the elements $z_1,\dots,z_m$, so
$I:X=I+J$ by (3.2).  Therefore it remains to show that this intersection is
irredundant.

For this, suppose, on the contrary, that it is redundant.  Then by resubscripting the $z_j$, if necessary, it may be assumed that $I=\cap\{P(z_j); j=1,\dots,k\}$ for some $k<m$ (so $z_1,\dots,z_k$ are the $I$ -corner-elements, by (4.8)).
Then $z_m\notin I$, since $z_m\notin P(z_m)\supseteq I$, so (3.15) shows
that there exists a monomial $g$ such that $gz_m$ is an $I$ -corner-element.
Therefore
$gz_m=z_i$ for some $i=1,\dots,k$, and this contradicts the hypothesis that
$z_i\notin z_jR$ for $i\not= j\in\{1,\dots,m\}$.  Therefore the intersection
is irredundant, \qed
\enddemo
\medskip
In (4.10) we show that a decomposition as in (4.1) of a monomial ideal is
unique.
\medskip
\proclaim{(4.10) Theorem (Unique Factorization)}
Let $z_1,\dots,z_m$ and $w_1,\dots,w_n$ be monomials such that
$\cap \{P(z_j); j=1,\dots,m\}=\cap \{P(w_i); i=1,\dots,n\}$ are irredundant
intersections of parameter ideals.  Then $n=m$ and $\{z_1,\dots,z_m\}=
\{w_1,\dots,w_n\}$.
\endproclaim

\demo{Proof}
Let $I=\cap \{P(z_j); j=1,\dots,m\}$.  Then it follows from (4.8) that
$z_1,\dots,z_m$ are the $I$ -corner-elements, so they are the monomials in $(I:X)-I$ by
(3.2).  However, $I=\cap\{P(w_i); i=1,\dots,n\}$, by hypothesis, so similar
statements hold for $w_1,\dots,w_n$ in place of $z_1,\dots,z_m$, hence it follows that $n$ $=$ $m$ and that $\{w_1,\dots,w_n\}$ $=$ $\{z_1,\dots,z_m\}$, \qed
\enddemo
\medskip
In (4.11) we note two additional results concerning $I$ -corner-elements.
\medskip
\proclaim{(4.11) Proposition}
Assume that  $x_1,\dots,x_d$ is a permutable  $R$ -sequence, let $I$ be
a monomial ideal such that $Rad(I)$ $=$ $Rad(X)$, let $z_1,\dots,z_m$ be the
$I$ -corner\linebreak
-elements, and let $f$ be a monomial.  Then:

{\pmb{(4.11.1)}} $fI=(\cap\{P(fz_j); j=1,\dots, m\})\cap fR$ and
$fz_1,\dots,fz_m$ are $fI$ -corner-elements.

{\pmb{(4.11.2)}} $I:fR=\cap\{P(w_j); j=1,\dots,k\}$, where $z_j=w_jf$ for $j=1,\dots, k$ and
$z_j\notin fR$ for $j=k+1,\dots,m$ (for some $k\in \{0,1,\dots,m\}$) and
$w_1,\dots,w_k$ are the $I:fR$ -corner-elements.
\endproclaim

\demo{Proof}
For (4.11.1), since each permutation of $x_1,\dots,x_d$ is an $R$ -sequence,
each monomial is regular.  Therefore since $z_j\in (I:X)-I$, by (3.2), it is
readily checked that $fz_j\in (fI:X)-fI$ for $j=1,\dots,m$, so each $fz_j$ is an
$fI$\linebreak
-\CE by (3.2).  Also, $P(fz_j):fR$ $=$ $P(z_j)$, for if $f$
$=$ $x_1^{b_1}\cdots x_d^{b_d}$  and  $z_j$ $=$
$x_1^{a_1}\cdots x_d^{a_d}$, then $P(fz_j):fR$ $=$
$(x_1^{a_1+b_1+1},\dots,x_d^{a_d+b_d+1})R:x_1^{b_1}\cdots x_d^{b_d}R$ $=$
$(x_1^{a_1+1},x_2^{a_2+b_2+1},\dots,x_d^{a_d+b_d+1})R:x_2^{b_2}\cdots x_d^{b_d}R$
$=\cdots=$
$(x_1^{a_1+1},\dots,x_d^{a_d+1})R$ $=$ $P(z_j)$.  Therefore it follows that
$(\cap \{P(fz_j); j=1,\dots,m\})\cap fR=f[(\cap \{P(fz_j); j=1,\dots,m\}):fR]=f[\cap \{P(fz_j):fR; j=1,\dots,m\}]=
f[\cap \{P(z_j); j=1,\dots,m\}]=fI$, the last equality by (4.1).

For (4.11.2), $z_j\in fR$ if and only if $f\notin P(z_j)$, by (2.3).  Therefore
if $z_j=fw_j$ for $j=1,\dots,k$ and if $z_j\notin fR$ (so $f\in P(z_j)$) for
$j=k+1,\dots,m$, then it follows from (4.1) that $I:fR=(\cap \{P(z_j);
j=1,\dots,m\}):fR=\cap \{P(z_j):fR; j=1,\dots,m\}=\cap \{P(w_jf):fR; j=1,\dots,k\}= \cap \{P(w_j); j=1,\dots,k\}$.  Finally, it follows from (4.8) that if $\cap\{P(w_j); j=1,\dots,k\}$ is an irredundant intersection, then
$w_1,\dots,w_k$ are the $I:fR$ -corner-elements, so it remains to show that this intersection is irredundant.

For this, suppose that the intersection is redundant, so (by resubscripting,
if necessary) there exists $h<k$ such that $I:fR=\cap\{P(w_j);j=1,\dots,h\}$
is an irredundant intersection, so $w_1,\dots,w_h$ are the
$I:fR$ -\CES by (4.8).  Therefore either (a) $w_k\in I:fR$;
or, (b) $w_k\notin I:fR$.  If (b) holds,
then $gw_k$ is an $I:fR$ -corner-element for some monomial $g$ by
(3.15), so $gw_k=w_j$ for some $j=1,\dots,h$ (so $g\not= 1$).  Therefore
$gz_k=fgw_k=fw_j=z_j$, so $z_j=gz_k\in I$ (by the definition of $I$ -corner-element,
since $g\not= 1$ is a monomial), and this contradicts the fact that $z_j$ in
an $I$ -corner-element.  Therefore (b) does not hold, so (a) holds, hence
$z_k=fw_k\in I$, and this contradicts the fact that $z_k$ is an $I$ -corner-element.
Therefore neither (a) nor (b) holds, so it follows that
$\cap \{P(w_j); j=1,\dots,k\}$ is an irredundant intersection, hence
$w_1,\dots,w_k$ are the $I:fR$ -corner-elements, \qed
\enddemo
\medskip
\proclaim{(4.12) Corollary}
With the notation of (4.11), let $J=(f_1,\dots,f_n)R$ be a monomial ideal and
let $w_{j,i}$ be monomials such that $z_j=w_{j,i}f_i$ (if $z_j\in f_iR$) or
$w_{j,i}=1$ (if $z_j\notin f_iR$).  Then $I:J=\cap \{P(w_{j,i}); j=1,\dots,m
\text{ and } i=1,\dots,n\}$, so the $I:J$ -\CES are among the $mn$ monomials
$w_{j,i}$.
\endproclaim

\demo{Proof}
If $w_{j,i}=1$, then $P(w_{j,i})=X$, and $X$ contains all other parameter
ideals.  Therefore the conclusion follows from (4.11.2) and the fact that
$I:J$ $=$ $\cap \{I: f_iR$; $i$ $=$ $1,\dots,n\}$, \qed
\enddemo
\medskip
To prove the next theorem, which gives an irredundant parametric decomposition
of the ideal generated by the $k$-th powers of the monomial
generators of a monomial
ideal, we need the following definition and lemma.
\medskip
\noindent
{\bf{(4.13) Definition.}}
If $J=(f_1,\dots,f_n)R$ is a monomial ideal and $k$ is a positive integer,
then $\bold J^{[k]}$ denotes the ideal $(f_1^k,f_2^k,\dots,f_n^k)R$.
\medskip
\proclaim{(4.14) Lemma}
Let $J$ be a monomial ideal, let $g$ be a monomial, and let $k$ be a positive
integer.  Then $g\in J$ if and only if $g^k\in J^{[k]}$.
\endproclaim

\demo{Proof}
It is clear that $g\in J$ implies that $g^k\in J^{[k]}$.

For the converse assume that $g^k\in J^{[k]}$.  Let $J=(f_1,\dots,f_n)R$, where
each $f_i$ is a monomial.  Then the hypothesis and (2.2.1) imply that
$g^k\in f_i^kR$ for some $i=1,\dots,n$.  Now $g^k$ and $f_i^k$ are monomials
in the $R$ -sequence $x_1^k,\dots,x_d^k$, so by (2.2.2) there exists a monomial
$s$ in $x_1^k,\dots,x_d^k$ such that $g^k=sf_i^k$.  Then it is clear that there
exists a monomial $t$ in $x_1,\dots,x_d$ such that $t^k=s$, so $g^k=t^kf_i^k$,
hence $g=tf_i\in J$, as desired, \qed
\enddemo
\medskip
\proclaim{(4.15) Theorem}
Let $I$ be a monomial ideal such that $Rad(I)$ $=$ $Rad(X)$,
let $z_1,\dots, z_m$ be
the $I$ -corner-elements, and let $k$ be a positive integer.  Then $I^{[k]}=
\cap\{(P(z_j))^{[k]}; j=1,\dots,m\}$ is a decomposition of $I^{[k]}$ as an
irredundant intersection of parameter ideals.
\endproclaim

\demo{Proof}
Let $I=(f_1,\dots,f_n)R$ and note that $x_1^k,\dots,x_d^k$ is an $R$ -sequence.
Therefore since each $f_i$ is a monomial (in $x_1,\dots,x_d$) and since
$I^{[k]}=(f_1^k,\dots,f_n^k)R$, it follows that $I^{[k]}$ is generated by
monomials in $x_1^k,\dots,x_d^k$, and $Rad(I^{[k]})$ $=$ $Rad(X^{[k]})$ (since
$Rad(I)$ $=$ $Rad(X)$).  Also, for each $j=1,\dots,m$ it holds that $z_j$ is a
monomial in $x_1,\dots,x_d$ such that $z_j\notin I$ and $z_jx_i\in I$ for
$i=1,\dots,d$, so it follows that $z_j^k$ is a monomial in $x_1^k,\dots,x_d^k$
such that $z_j^k\notin I^{[k]}$ (by (4.14)) and $z_j^kx_i^k\in I^{[k]}$ for
$i=1,\dots,d$.  Therefore the $m$ elements $z_1^k,\dots,z_m^k$ are among the
$I^{[k]}$ -\CES (for the $R$ -sequence $x_1^k,\dots,x_d^k$).

Now let $z^*$ be an $I^{[k]}$ -\CE (for the $R$ -sequence $x_1^k,\dots,x_d^k$),
so $z^*$ is a monomial in $x_1^k,\dots,x_d^k$ such that $z^*\notin I^{[k]}$
and $z^*x_i^k\in I^{[k]}$ for $i=1,\dots,d$.  Then it is clear that there
exists a monomial $z$ in $x_1,\dots, x_d$ such that $z^k=z^*$, so $z\notin I$
(since $z^k\notin I^{[k]}$) and $zx_i\in I$ (by (4.14), since
$z^kx_i^k\in I^{[k]}$).  Therefore $z$ is an  $I$ -corner-element, so $z=z_p$ for some
$p=1,\dots,m$, so $z^*=z^k=z_p^k$.  Therefore it follows that
$z_1^k,\dots,z_m^k$ are all the $I^{[k]}$ -\CES (for $x_1^k,\dots,x_d^k$) so it
follows from (4.1) that $I^{[k]}=\cap \{P(z_j^k); j=1,\dots,m\}$.

Finally, fix $j\in \{1,\dots,m\}$ and let $z_j=x_1^{a_1}\cdots x_d^{a_d}$.  Then
$z_j^k=x_1^{ka_1}\cdots x_d^{ka_d}=(x_1^k)^{a_1}\cdots (x_d^k)^{a_d}$, so it
follows from (2.1.2) that $P(z_j^k)=((x_1^k)^{a_1+1},\dots,(x_d^k)^{a_d+1})R$ and that $P(z_j)=(x_1^{a_1+1},\dots,x_d^{a_d+1})R$, so it follows that
$P(z_j^k)=(P(z_j))^{[k]}$.  Therefore it follows from the preceding paragraph
that $I^{[k]}=\cap \{(P(z_j))^{[k]}); j=1,\dots,m\}$, \qed
\enddemo
\medskip
The final result in this section shows that the  $I$ -corner-elements
determine the $I^{[k]}$ -corner-elements.
\medskip
\proclaim{(4.16) Corollary}
With the notation of (4.15), $c(I)=c(I^{[k]})$ (see (3.9)).  More specifically,
if $z_1,\dots,z_m$ are the $I$ -corner-elements, and if $z_j=x_1^{a_{j,1}}\cdots x_d^{a_{j,d}}$, then the $I^{[k]}$ -\CES are the $m$ monomials $z_j^{(k)}=
x_1^{ka_{j,1}+k-1}\cdots x_d^{ka_{j,d}+k-1}$.
\endproclaim

\demo{Proof}
This follows immediately from (4.15) and (4.8), \qed
\enddemo
\bigskip
\noindent
{\smc{\bf 5.  A Related Irredundant Irreducible Decomposition.}}
Let $I$ be a monomial ideal in a local ring $(R,M)$ such that $Rad(I)$
$=$ $Rad(X)$.
Then the main result in this section, (5.1), gives a decomposition of
$I+M(I:X)$ that is closely related to (4.1).
\medskip
\proclaim{(5.1) Theorem}
Assume that $R$ is local with maximal ideal $M$, let $I$ be a monomial ideal
such that $Rad(I)$ $=$ $Rad(X)$, let $z_1,\dots,z_m$ be the $I$ -corner-elements, and for
$j=1,\dots,m$ let $Q_j$ be maximal in $S_j=\{Q;Q \text{ is an ideal in } R,
P(z_j)\subseteq Q$, and $z_j\notin Q\}$.  Then each $Q_j$ is irreducible,
$\underset{j=1}\to{\overset m \to \cap}\ Q_j$ is an irredundant intersection,
and $(\underset{j=1}\to{\overset m \to \cap}\ Q_j)\cap (I:X)=I+M(I:X)$.
\endproclaim

\demo{Proof}
Fix $j\in \{1,\dots,m\}$.  Then $P(z_j)\in S_j$, by (2.3), so $S_j$ is not empty, so there exists an ideal $Q_j$ that is maximal with respect to being in
$S_j$.  Then each ideal that properly contains $Q_j$ must contain $z_j$, so
$Q_j$ is irreducible.

Also, $z_j \in P(z_i)\subseteq Q_i$ for $i\in \{1,\dots,j-1,j+1,\dots,m\}$
(by (3.3)) and $z_j\notin Q_j$, so $\underset{j=1}\to{\overset m \to \cap}\ Q_j$
is an irredundant intersection.

Further, since $z_jM\subset z_jR$, it follows that $z_jM\subseteq Q_j$, and
$I$ $+$ $(z_1,\dots,z_{j-1},$\linebreak
$z_{j+1},\dots z_m)R$
$\subseteq$ $P(z_j)$
$\subseteq$ $Q_j$, by (3.3) and (4.1),
so it follows from (3.7) that $I$
$+$ $M(I:X)$ $\subseteq$ $I$ $+$ $(z_1,\dots,z_{j-1},z_{j+1},\dots z_m)R$
$+$ $z_jM$ $\subseteq$ $Q_j$.  Therefore it
follows that $I+M(I:X)$ $\subseteq$
$(\underset{j=1}\to{\overset m \to \cap}\ Q_j) \cap (I:M)$.

Finally, if $y\in (\underset{j=1}\to{\overset m \to \cap}\ Q_j)\cap (I:M)$,
then $y=\underset{i=1}\to{\overset n \to\sum}\, s_if_i+\underset{j=1}\to{\overset m \to\sum}\,r_jz_j$ (by (3.7), where $I=(f_1,\dots,f_n)R$) and $y$ is in each
$Q_j$.  However, since $I+(z_1,\dots,z_{j-1},z_{j+1},\dots,z_m)R+z_jM\subseteq Q_j$ and $z_j\notin Q_j$, it follows that $r_jz_j\in Q_j$, hence
$r_j\in Q_j:z_jR$ $=$ $M$.  Since this holds for each $j=1,\dots,m$ it follows
that $y\in I+M(I:X)$, \qed
\enddemo
\medskip
\noindent
{\bf{(5.2) Remark.}}
It is readily checked that the following are equivalent for two ideals $J$
and $Y$ in a local ring $(R,M)$:  (a) $M(J:Y)\subseteq J$;  (b) $J:Y=J:M$;
(c) $J:(J:Y)=M$.  If any of (a) - (c) hold with $I$ and $X$ in place of $J$
and $Y$, then $X=M$ and $R$ is a regular local ring.

\demo{Proof}
It follows from [T, Theorem 6] that $I:(I:X)$ is generated by monomials, so
if any of (a) - (c) hold, then in particular (c) holds, so $M$ is generated by
monomials.  But every ideal generated by monomials is contained in $X$, so
$M=X$ is generated by an $R$ -sequence, hence $R$ is a regular local ring, \qed
\enddemo
\medskip
\proclaim{(5.3) Corollary}
With the notation of (5.1) assume that $R$ is a regular local ring with
maximal ideal $M=X$.  Then $I=\cap \{Q_j; j=1,\dots,m\}$ is an irredundant
irreducible decomposition of $I$ and $Q_j=P(z_j)$ for $j=1,\dots,m$.
\endproclaim

\demo{Proof}
Since $X=M$, $M(I:X)\subseteq I$, so (5.1) shows that
$(\underset{j=1}\to{\overset m \to \cap}\ Q_j)\cap (I:M)=I$, hence
$(\underset{j=1}\to{\overset m \to \cap}\ (Q_j/I))\cap ((I:M)/I)=I/I$.  Also,
$(I:X)/I=(I:M)/I$ is the socle of $R/I$, and it is shown in [HRS3, (3.3.2)]
that $(\underset{j=1}\to{\overset m \to \cap}\ (Q_j/I))\cap ((I:M)/I)=(0)$ if and only if $\underset{j=1}\to{\overset m \to \cap}\ (Q_j/I)=(0)$.  It
therefore follows that $I=\cap \{Q_j; j=1,\dots,m\}$, and (5.1) shows that this
is an irredundant intersection of irreducible ideals.

To see that $Q_j=P(z_j)$ for $j=1,\dots,m$, fix $j\in \{1,\dots,m\}$ and
note that it is shown in (4.1) that $I=\cap \{P(z_j); j=1,\dots,m\}$.
Since $R$ is regular, it follows that each parameter ideal $P(z_j)$ is
irreducible.  Also, $P(z_j)\subseteq Q_j$, by construction (see (5.1)).
Further, it is shown in [HRS3, (3.6)] that there are no containment relations
among the ideals in $IC(I)$ $=$ $\{q$; there exists an irredundant
irreducible decomposition of  $I$  with  $q$  as a factor$\}$.  Therefore
it follows that
$Q_j=P(z_j)$ for $j=1,\dots,m$, \qed
\enddemo
\medskip
\noindent
{\bf{(5.4) Remark.}}
If either $M\not= X$ or if $R$ is a Gorenstein local ring, but not regular,
in (5.3), then the parameter ideal $P(z_j)$ is irreducible and every monomial
ideal that contains $P(z_j)$ must contain $z_j$ (by (3.16), since $z_j$
is the unique $P(z_j)$\linebreak
-corner-element, by (4.3)).  However, there are ideals that contain
$P(z_j)$ that are not monomial ideals, so the unique cover of $P(z_j)$ is
properly contained in $(P(z_j),z_j)R$, and hence $P(z_j)\subset Q_j$ in (5.1).
\bigskip
\noindent
{\smc{\bf 6. Parametric and Irreducible ideals.}}  In this section we prove
a few additional results concerning parameter ideals and their relation to
irreducible ideals.
\medskip
\proclaim{(6.1) Proposition}
Consider the following statements about a monomial ideal $I$ such that
$Rad(I)$ $=$ $Rad(X)$:

{\pmb{(6.1.1)}}\quad $I$ has exactly one corner-element.

{\pmb{(6.1.2)}}\quad $I$ is a parameter ideal.

{\pmb{(6.1.3)}}\quad $I$ is irreducible.

Then (6.1.3) $\Rightarrow$ (6.1.1) $\Leftrightarrow$ (6.1.2), and all three
statements are equivalent when  $R$  is Cohen-Macaulay and  $Rad(X)$ $=$
$P$  is a prime ideal such that  $R_P$  is a Gorenstein
local ring of altitude $d$.
\endproclaim

\demo{Proof}
Since $Rad(I)$ $=$ $Rad(X)$, (4.1) shows that $I=\cap \{P(z_j); j=1,\dots,m\}$,
where $z_1,\dots,z_m$ are the $I$ -corner-elements.  Therefore if $I$ is irreducible, then
$m=1$, so (6.1.3) $\Rightarrow $(6.1.1).

(4.1) shows that (6.1.1) $\Rightarrow$ (6.1.2).

Assume that (6.1.2) holds and let $I=(x_1^{a_1},\dots,x_d^{a_d})R$.  Then
$x_1^{a_1-1}\cdots x_d^{a_d-1}$ is the unique $I$ -\CE (since
$I:X=x_1^{a_1-1}\cdots x_d^{a_d-1}R)$, so (6.1.2) $\Rightarrow$ (6.1.1).

Finally, if $R$ is Cohen-Macaulay and $Rad(X)$ $=$ $P$ is a prime ideal such that
$R_P$ is a Gorenstein local ring, then it readily follows from (4.3) that
(6.1.1) $\Rightarrow$ (6.1.3), \qed
\enddemo
\medskip
\noindent
{\bf{(6.2) Remark.}}
(6.1) provides an alternate proof that open monomial ideals are finite
intersections of parameter ideals in Gorenstein local rings.  Specifically,
let $Q$ be such an ideal.  If $Q$ is irreducible, then $Q$ is a parameter ideal
by (6.1).  If $Q$ is not irreducible, then $Q$ is the intersection of two
monomial ideals that properly contain it.  (For if $Q=(f_1,\dots,f_n)R$
and $f_i=x_1^{e_{i,1}}\cdots x_d^{e_{i,d}}$ is such that $e_{i,j}\ge 1$
and $e_{i,k}\ge 1$, then $Q=Q_1\cap Q_2$, where $Q_1=(f_1,\dots,f_{i-1},x_1^{e_{i,1}}\cdots x_j^{e_{i,j}-1}\cdots x_d^{e_{i,d}},
f_{i+1},\dots,f_n)R$ and $Q_2=(f_1,\dots,f_{i-1},x_1^{e_{i,1}}\cdots x_k^{e_{i,k}-1}\cdots x_d^{e_{i,d}},f_{i+1},\dots,f_n)R$.)  Therefore by induction on the (finite) number of monomial ideals between $X$ and $Q$ it
follows that the open monomial ideals $Q_1$ and $Q_2$ are finite intersections
of parameter ideals, so $Q$ is.
\medskip
(6.3) characterizes the parameter ideals that are minimal with respect to
containing a given monomial ideal $I$.
\medskip
\proclaim{(6.3) Proposition}
Let $I$ be a monomial ideal such that $Rad(I)$ $=$ $Rad(X)$ and let $Q$ be an ideal
that is minimal in $\{q; I\subseteq q \text{ and  $q$  is a parameter ideal}\}$.
Then $Q=P(z)$ for some $I$ -\CE $z$.
\endproclaim

\demo{Proof}
By (4.1) and (6.1), $	Q=P(w)$ for the unique $Q$ -\CE $w$.  Then $w\notin Q$,
so $w\notin I$, hence $fw$ is an $I$ -\CE for some monomial $f$ by (3.15).
Then $I\subseteq P(fw)\subseteq P(w)=Q$, and $P(fw)$ is a parameter ideal.
Therefore the definition of $Q$ shows that $P(fw)=P(w)$, so $w=fw$ is an
$I$ -corner-element, \qed
\enddemo
\medskip
\proclaim{(6.4) Corollary}
Let $I\subseteq J$ be monomial ideals such that $Rad(I)$ $=$ $Rad(X)$, and let
$z_1,\dots z_m$ (resp., $w_1,\dots,w_n)$ be the $I$ (resp. $J$) -corner-elements. Then
each 
$P(w_i)$ contains some $P(z_j)$ and then $z_j\in w_iR$.
\endproclaim

\demo{Proof}
Let $\bold P(I)=\{q; I\subseteq q \text{ and $q$ is a parameter ideal}\}$.
Then $I\subseteq J\subseteq P(w_i)$ for $i=1,\dots,n$, by (4.1), so each
$P(w_i)\in \bold P(I)$.  Fix $i\in \{1,\dots,n\}$.  Then $P(w_i)\in \bold P(I)$,
so $P(w_i)$ contains an ideal $q$ that is minimal in $\bold P(I)$.  Then
$q=P(z_j)$ for some $I$ -\CE $z_j$, by (6.3), so $P(z_j)\subseteq P(w_i)$, and
it is readily checked that this implies that $z_j\in w_iR$, \qed
\enddemo
\medskip
In our final result, by ``$Q$ is an irreducible component of $I$'' we mean that
there exists a decomposition $\cap \{Q_j; j=1,\dots,m\}$ of $I$ as an
irredundant finite intersection of irreducible ideals $Q_j$ such that
$Q=Q_j$ for some $j=1,\dots,m$.
\medskip
\proclaim{(6.5) Corollary}
Let $I$ be a monomial ideal such that $Rad(I)$ $=$ $Rad(X)$ and let $Q$ be minimal
in $\{q$; $I$ $\subseteq$ $q$  and  $q$  is an irreducible monomial ideal in $R\}$.  If $R$ is a Gorenstein local ring, then $Q$ is an irreducible
component of $I$.
\endproclaim

\demo{Proof}
If $R$ is a Gorenstein local ring, and if $Q$ is an irreducible monomial
ideal, then $Q$ is a parameter ideal, by (6.1), so this follows immediately
from (6.3) and (4.1), \qed
\enddemo
\bigskip
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\endRefs


\enddocument

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