Chicago, Chicago, Illinois, USA b Department of Mathematics NDSU, Fargo, North Dakota, USA

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1 This article was downloaded by: [Sean Sather-Wagstaff] On: 25 August 2013, At: 09:06 Publisher: Taylor & Francis Infora Ltd Registered in England and Wales Registered Nuber: Registered office: Mortier House, Mortier Street, London W1T 3JH, UK Counications in Algebra Publication details, including instructions for authors and subscription inforation: Decopositions of Monoial Ideals in Real Seigroup Rings Daniel Ingebretson a & Sean Sather-Wagstaff b a Departent of Matheatics, Statistics, and Coputer Science, University of Illinois at Chicago, Chicago, Illinois, USA b Departent of Matheatics NDSU, Fargo, North Dakota, USA To cite this article: Daniel Ingebretson & Sean Sather-Wagstaff (2013) Decopositions of Monoial Ideals in Real Seigroup Rings, Counications in Algebra, 41:11, To link to this article: PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis akes every effort to ensure the accuracy of all the inforation (the Content ) contained in the publications on our platfor. However, Taylor & Francis, our agents, and our licensors ake no representations or warranties whatsoever as to the accuracy, copleteness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with priary sources of inforation. Taylor and Francis shall not be liable for any losses, actions, clais, proceedings, deands, costs, expenses, daages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content. This article ay be used for research, teaching, and private study purposes. Any substantial or systeatic reproduction, redistribution, reselling, loan, sub-licensing, systeatic supply, or distribution in any for to anyone is expressly forbidden. Ters & Conditions of access and use can be found at

2 Counications in Algebra, 41: , 2013 Copyright Taylor & Francis Group, LLC ISSN: print/ online DOI: / DECOMPOSITIONS OF MONOMIAL IDEALS IN REAL SEMIGROUP RINGS Daniel Ingebretson 1 and Sean Sather-Wagstaff 2 1 Departent of Matheatics, Statistics, and Coputer Science, University of Illinois at Chicago, Chicago, Illinois, USA 2 Departent of Matheatics NDSU, Fargo, North Dakota, USA Key Words: Irreducible decopositions of onoial ideals in polynoial rings over a field are well-understood. In this article, we investigate decopositions in the set of onoial ideals in the seigroup ring A 0 d where A is an arbitrary coutative ring with identity. We classify the irreducible eleents of this set, which we call -irreducible, and we classify the eleents that adit decopositions into finite intersections of -irreducible ideals. Irreducible decopositions; Monoial ideals; Seigroup rings Matheatics Subject Classification: 13C05; 13F INTRODUCTION Throughout this article, let A be a coutative ring with identity. When A is a field, the polynoial ring P A X 1 X d is noetherian, so every ideal in this ring has an irreducible decoposition. For onoial ideals, that is, the ideals of P generated by sets of onoials, these decopositions are well understood: the non-zero irreducible onoial ideals are precisely the ideals X e 1 i 1 X e n i n P generated by pure powers of soe of the variables, and every onoial ideal in this setting decoposes as a finite intersection of irreducible onoial ideals. Furtherore, there are good algoriths for coputing these decopositions, both by hand [6, 11, 12, 16, 24] and by coputer [2, 7, 8]. 1 Note that uch of this discussion extends to the case where P is replaced by a nuerical seigroup ring, that is, a ring of the for A S where S is a sub-seigroup of d. When A is not noetherian, soe of the conclusions fro the previous paragraph fail because P fails to be noetherian. However, P does behave soewhat noetherianly with respect to onoial ideals. For instance, all onoial ideals Received August 13, 2011; Revised May 18, Counicated by I. Swanson. Address correspondence to Sean Sather-Wagstaff, Departent of Matheatics, NDSU Dept. #2750, P.O. Box 6050, Fargo, ND , USA; E-ail: sean.sather-wagstaff@ndsu.edu 1 One of the ost interesting aspects of this theory is found in its interactions with cobinatorics, including applications to graphs and siplicial coplexes; see, e.g., [3 5, 9, 10, 15, 17 19, 21, 22, 25, 27]. Foundational aterial on the subject can be found in the following texts [1, 13, 14, 20, 23, 26, 28]. 4363

3 4364 INGEBRETSON AND SATHER-WAGSTAFF in P are finitely generated, by finite sets of onoials. The ideals that are indecoposable with respect to intersections of onoial ideals, which we call -irreducible, are the ideals X e 1 i 1 X e n i n P. Each onoial ideal of P adits an -irreducible decoposition, a decoposition into a finite intersection of -irreducible onoial ideals, and any of the algoriths carry over to this setting; see, e.g., [23]. In this article, we step even further fro the noetherian setting by considering onoial ideals in the seigroup ring R A 0 d where A is an arbitrary coutative ring with identity and 0 d r 1 r d d r 1 r d 0. This ring can be thought of as the set A X 0 1 X 0 d of all polynoials in variables X 1 X d with coefficients in A where the exponents are non-negative real nubers. For instance, in this ring, any of the onoial ideals are not finitely generated and any do not adit finite -irreducible decopositions; see, e.g., Fact 3.3(c) and Exaple On the other hand, every ideal adits a (possibly infinite) -irreducible decoposition, by Proposition Our goal is to copletely characterize the onoial ideals in R that adit - irreducible decopositions. This is accoplished in two steps. First, we characterize the -irreducible ideals. This is accoplished in Theore 3.9, which we paraphrase in the following result. Theore 1.1. A onoial ideal in the ring A 0 d A X 0 1 X 0 d is -irreducible if and only if it is generated by a set of pure powers of the variables X 1 X d. Our characterization of the onoial ideals in R that adit -irreducible decopositions is ore technical. However, our intuition is straightforward, and reflects the connection between the noetherian property and existence of decopositions: a onoial ideal in R adits an -irreducible decoposition if and only if it is alost finitely generated. To ake sense of this, we need to explain what we ean by alost finitely generated. We build up the general definition in steps. First, we consider the case of onoial ideals that are alost principal. In one variable (i.e., the case d 1) there are exactly two kinds of nonzero onoial ideals: given a real nuber a 0 set I a 0 X r 1 r a R Xa 1 R I a 1 X r 1 r > a R These ideals are copletely deterined by the sets of their exponents, corresponding exactly to open and closed rays in 0. We think of these ideals as being alost generated by X1 a. This includes the ideal that is generated by Xa 1 as a special case. In two variables (i.e., the case d 2) there is ore variation. First, not every ideal is alost principal; in fact, we can find ideals here that are not alost finitely generated here. Second, the alost principal ideals coe in four flavors in this setting: given real nubers a b 0 set I a b 0 0 X r 1 Xs 2 r a and s b R Xa 1 Xb 2 R I a b 1 0 X r 1 Xs 2 r>aand s b R

4 DECOMPOSITIONS OF MONOMIAL IDEALS 4365 I a b 0 1 X r 1 Xs 2 r a and s > b R I a b 1 1 X r 1 Xs 2 r>aand s > b R These ideals are copletely deterined by the sets of their exponent vectors, corresponding to cobinations of open and closed rays on each axis of 2.We think of these ideals as being alost generated by X1 axb 2. In general (i.e., for arbitrary d 1) there are 2 d different flavors of alost principal onoial ideals, corresponding to the different choices of open and closed rays for the exponents of each variable; see Notation 4.2. In general, a onoial ideal is alost finitely generated if it is a finite su of alost principal onoial ideals. This definition is otivated by the fact that a finitely generated onoial ideal is a su of principal onoial ideals. For instance, each -irreducible onoial ideal is alost finitely generated. In these ters, our characterization of the decoposable onoial ideals, stated next, is quite straightforward; see Theore 4.12: Theore 1.2. A onoial ideal in R adits an -irreducible decoposition if and only if it is alost finitely generated. As to the organization of this article, Section 2 consists of definitions and background results, and Sections 3 and 4 are priarily concerned with the proofs of Theores 1.1 and 1.2, respectively. 2. BACKGROUND AND PRELIMINARY RESULTS In this section, we lay the foundation for the proofs of our ain results. We begin by establishing soe notation for use throughout the article. Definition 2.1. Set 0 r r 0 Let d be a non-negative integer, and set 0 0 d 0 0 d { r 1 r d d r 1 r d 0 } which is an additive seigroup, and d 0 0 d { r 1 r d r 1 r d 0 } We consider the seigroup ring R A d 0 which we think of as the set of all polynoials in variables X 1 X d with coefficients in A where the exponents are non-negative real nubers. Moreover, for i 1 d,weset X i 0

5 4366 INGEBRETSON AND SATHER-WAGSTAFF When the nuber of variables is sall (d 2), we will use variables X Y in place of X 1 X 2.Aonoial in R is an eleent of the for X r X r 1 1 Xr d d R where r r 1 r d 0 d is the exponent vector of the onoial f. Multiplication of onoials in this ring is defined analogously to its classical counterpart. For all q r 0 d, and all s, we write X q X r X q+r X q s X sq An arbitrary eleent of R is a linear cobination of onoials f finite r d 0 a r X r with coefficients a r A. Aonoial ideal of R is an ideal I S R generated by a set S of onoials in R. Given a subset G R, the onoial set of G is G onoials of R in G G We next list soe basic properties of onoial ideals. Fact 2.2. Let I and J be onoial ideals of R. (a) For any subset G R, we have G G R, by definition. (b) The onoial ideal I is generated by its onoial set: I I R. (c) Parts (a) and (b) cobine to show that I J if and only if I J, and hence I J if and only if I J. The next facts follow fro the 0 d -graded structure on R, that is, the isoorphiss R d 0 AX r f R Af. Fact 2.3. Let I be a set of onoial ideals of R. (a) Given onoials f X r and g X s in R, we have f g if and only if g f R if and only if s i r i for all i. When these conditions are satisfied, we have g fh where h X s r. (b) Given a onoial f R and a subset S R, we have f S R if and only if f s R for soe s S. (c) The su I is a onoial ideal such that I I. (d) The intersection I is a onoial ideal such that I I. Given a set S of subsets of R, the equality S R S R is standard. In general, one does not have such a nice description for the intersection S R. However, for onoial ideals in our ring R, the next result provides such a description. In a sense, it says that the onoial ideals of R behave like ideals in a unique factorization doain. First, we need a definition.

6 DECOMPOSITIONS OF MONOMIAL IDEALS 4367 Definition 2.4. Let X r 1 X r k R with r i r i 1 r i d 0 d. We define the least coon ultiple of these onoials as lc 1 i k Xr k X p where p is defined coponentwise by p j ax 1 i k r i j. Lea 2.5. Given subsets S 1 S k R, we have k S i R i1 ( ) lc f i f i S i for i 1 k R 1 i k Proof. Let L lc 1 i k f i f i S i for i 1 k R, which is a onoial ideal of R by definition. Fact 2.3(d) iplies that k i1 S i R is also a onoial ideal with k i1 S i R k i1 S i R. Thus, to show that k i1 S i R L, we need only show that k i1 S i R L. For the containent k i1 S i R L, let X t k i1 S i R. Fact 2.3(b) iplies that X t is a ultiple of one of the generators X r i S i for i 1 k. Hence we have t j r i j for all i and j by Fact 2.3(a). It follows that t j ax 1 i d r i j p j, hence X t X p R L. For the reverse containent k i1 S i R L, let X t L. Fact 2.3(b) iplies that X t is a ultiple of one of the onoial generators of L, so there exist X r i S i for i 1 k such that X t is a ultiple of lc 1 i k X r i. Since lc 1 i k X r i is a ultiple of X r i for each i, we then have X t k i1 Xr i R k i1 S i R, as desired. Lea 2.6. Let G R, and set I G R. Let X b I be given, and for j 1 d set I j G X b j j R. Then we have I d j1 I j. Proof. The containent G G X b j j iplies that I G R G X b j j R I j, so we have I d j1 I j. For the reverse containent, Facts 2.2(c) and 2.3(d) iply that it suffices to show that I d j1 I j. Let d j1 I j, and suppose that I. For j 1 d, we have I j G X b j j R. The condition I G R iplies that is not a ultiple of any eleent of G, so Fact 2.3(c) iplies that X b j j R for j 1 d. In other words, we have d j1 Xb j j R X b R I by Lea 2.5. This contradiction establishes the lea. The last result of this section describes the interaction between sus and intersections of onoial ideals in R. Lea 2.7. For t 1 l, let K t it t i t 1 be a collection of onoial ideals. Then the following equalities hold: l t 1 2 l l (a) K t it K t it t1 i t 1 i 1 1 i 2 1 i l 1 t1

7 4368 INGEBRETSON AND SATHER-WAGSTAFF (b) l t t1 i t 1 K t it 1 2 i 1 1 i 2 1 l i l 1 t1 l K t it Proof. (a) Note that the left- and right-hand sides of Equation (a) are onoial ideals of R by Fact 2.3(c) (d). Because of Fact 2.2(c), Equation (a) follows fro the next sequence of equalities: [[ ]] [[ ]] l t l t K t it K t it t1 i t 1 t1 l t1 1 i t 1 [ t i t 1 2 i 1 1 i i 1 1 i 2 1 [[ 1 [[ Kt it ]] ] 2 i 1 1 i 2 1 l i l 1 l i l 1 [ l [[ ]] ] Kt it t1 [[[ ]]] l K t it t1 ]] l l K t it i l 1 t1 Here, the third equality is fro the distributive law for unions and intersections, while the reaining steps are fro Fact 2.3(c) (c). The verification of Eq. (b) is siilar, so we oit it. 3. M-IRREDUCIBLE IDEALS The following notation is extreely convenient for our proofs. To otivate the notation, note that when is 1, we are thinking of as an arbitrarily sall positive real nuber. Notation 3.1. Let 2. Given r, we define { r if 0 r provided that r> if 1 Given s 0, we define s provided that s Eploying this new notation, we define a onoial ideal J i that is generated by pure powers of the single variable X i. Recall our convention that X i 0. Notation 3.2. Given 0 and 2,weset J i X r i r R We use the ter pure power to describe a onoial of the for X r i.

8 DECOMPOSITIONS OF MONOMIAL IDEALS 4369 Given 0, we use 2 to distinguish between two iportant cases. Essentially, they represent the difference between the closed interval in the case 0 and the open interval in the case 1. The iportant difference is the existence of a inial eleent in the first case, but not in the second case. The case ay see strange, but it is quite useful. Fact 3.3. Let 0 and 2. (a) J i 0. (b) If 0, then J i Xi R. (c) If <, then J i is finitely generated if and only if 0. The ideal J i is generated by pure powers of the single variable X i. Next, we consider the class of ideals generated by pure powers of ore than one variable. This notation is the first place where we see the utility of our convention X i 0, since it allows us to consider all the variables siultaneously instead of worrying about partial lists of the variables. Notation 3.4. Given 1 d d 0, and 1 d d 2,weset J ({ X r i i i 1 d and r i i i }) R Exaple 3.5. In the case d 2, using constants a b 0 and 2,we have eight different possibilities for J : J a b 0 0 X a Y b R J a b 1 0 X a Y b a > a R J b 0 Y b R J a 1 X a a > a R J a b 0 1 X a Y b b > b R J a b 1 1 X a Y b a >aand b > b R J b 1 Y b b > b R J 0 The following connection between ideals of the for J and those of the for J i is iediate. Fact 3.6. Given 0 d and d 2, we have the following equality: J d J i i i i1 The ideals defined next are the irreducible eleents of the set of onoial ideals. Definition 3.7. A onoial ideal I R is -irreducible (short for onoialirreducible) provided that for all onoial ideals J and K of R such that I J K, either I J or I K. A straightforward induction arguent establishes the following property.

9 4370 INGEBRETSON AND SATHER-WAGSTAFF Fact 3.8. Let I be an -irreducible onoial ideal of R. Given onoial ideals I 1 I n of R, ifi n j1 I j, then I I j for soe j. Our first ain result, which we prove next, is the fact that the -irreducible onoial ideals of R are exactly the J ; it contains Theore 1.1 fro the introduction. Notice that it includes the case J 0, where. Theore 3.9. Let I R be a onoial ideal. Then the following are equivalent: (i) I is generated by pure powers of a subset of the variables X 1 X d ; (ii) There exist d 0 and d 2 such that I J ; and (iii) I is -irreducible. Proof. (i) (ii) Assue that I is generated by pure powers of a subset of the variables. It follows that there are (possibly epty) sets S 1 S d 0 such that I d i1 Xz i i z i S i R. For i 1 d set i inf S i 0, and let 1 d 0 d. (Here we assue that inf.) Furtherore, for i 1 d set { 0 if i i S i 1 if i S i and let 1 d d 2. It is straightforward to show that I J. (ii) (iii) Let 0 d and d 2 be given, and suppose by way of contradiction that J is not -irreducible. By definition, there exist onoial ideals J and K such that J J K, with J J and J K. It follows that J J and J K, so Fact 2.2(c) provides onoials X q J\J and X r K\J. If there exist q i or r i such that r i i i or q i i i, then Fact 2.3(a) iplies that X q X i i J, a contradition. We conclude that for all i, we have q i < i and r i < i,sop i ax q i r i < i. By Fact 2.3(a), this iplies that X p lc X q X r is not a ultiple of any generator of J, hence X p J by Fact 2.3(b). However, Lea 2.5 iplies that X p J K J, a contradiction. (iii) (i) Assue that I is -irreducible, and let G R be a generating set for I. Suppose by way of contradiction that I is not generated by pure powers of soe of the variables X 1 X d. Then there is a onoial X b G that is not a ultiple of any pure power Xi a G. For j 1 d, we consider the onoial ideals I j G X b j j R. Clai: I I j for j 1 d. Suppose by way of contradiction that there exists an index k such that I I k, that is, such that G R G X b k k R. Fact 2.2(c) iplies that X b k k is a ultiple of soe g G. By Fact 2.3(a), we conclude that g Xk a for soe a b k, so we have X b X b k k R Xa k R. That is, the onoial Xb is a ultiple of the pure power Xk a g G, a contradiction. This establishes the clai. Lea 2.6 iplies that I d j1 I j. Thus, the clai conspires with Fact 3.8 to contradict the fact that I is -irreducible.

10 DECOMPOSITIONS OF MONOMIAL IDEALS 4371 We explicitly docuent a special case of Theore 3.9 for use in the sequel. Corollary Each ideal J i is -irreducible. Proof. This is the special case of Theore IDEALS ADMITTING FINITE M-IRREDUCIBLE DECOMPOSITIONS We now turn our attention to the task of characterizing the onoial ideals of R that adit decopositions into finite intersections of -irreducible ideals. Definition 4.1. Let I R be a onoial ideal. An -irreducible decoposition of I is a decoposition I I where each I is an -irreducible onoial ideal of R. If the index set is finite, we say that I I is a finite -irreducible decoposition. Our second ain result shows that the onoial ideals of R that adit finite -irreducible decopositions are precisely the finite sus of ideals of the next for. Notation 4.2. Let 0 d and d 2 be given, and set I ({ }) X r i 1 d and r i i i R Exaple 4.3. With the zero-vector 0 0 0, we have I 0 X R. If i for any i, then I 0. As a first step, we show next that each ideal of the for I has a finite - irreducible decoposition. Lea 4.4. Given d 0 and d 2, we have I d i1 J i i i. Proof. If i for soe i, then we have J i i i 0, so d i1 J i i i 0; thus the desired equality follows fro Exaple 4.3 in this case. Assue now that i for all i. In the following coputation, the second equality is fro Lea 2.5: d d ( r J i i i X i i r i i i ) R i1 i1 ( ) { r lc X i } i ri i i R 1 i d ({ X r 1 1 Xr 2 2 Xr d d r }) i i i R ({ }) X r r i i i R I The first, fourth, and fifth equalities are by definition, and the third equality is straightforward.

11 4372 INGEBRETSON AND SATHER-WAGSTAFF Reark 4.5. It is worth noting that the decoposition fro Lea 4.4 ay be redundant, in the sense that soe of the ideals in the intersection ay be reoved without affecting the intersection: if i 0 and i 0, then J i i i Xi 0 R R. Lea 4.4 not only provides a decoposition for the ideal I, but also gives a first indication of how the ideals J i i i behave under intersections. The next leas partially extend this. The essential point of the first lea is the fact that a finite intersection of real intervals of the for i and j is a real interval of the for or. Lea 4.6. Let i b be given such that 1 i d and b 0. Given 1 b 0 and 1 b 2, there exist 0 and 2 such that b t1 J i t t J i. Specifically, we have { ax 1 b if b 1 0 if b 0 Proof. If b 0, then we have b 0 J i t t t1 J i t t t1 R J i 0 0 as claied. Thus, we assue for the reainder of the proof that b 1. Since b t1 J i t t J i j j for each j, it suffices to find an index j such that j ax 1 b and J i j j J i t t for all t. If j for soe j, then we have J i j j 0, and we are done. Thus, we assue for the reainder of the proof that j for all j. Choose k such that k ax 1 b. If there is an index j such that j k and j 1, then we have J i j j J i t t for all t since J i j j is generated by onoials of the for Xi where > j t. So, we assue that for every index j such that j k, we have j 0. In this case, we have J i k k J i t t for all t, as follows. If k t, then t 0, so we have J i k k X k i R X t i R J i t t. On the other hand, if k t, then k > t, and hence J i k k X k i R J i t t. Lea 4.7. Let k be a positive integer. For t 1 k let i t 1 d be given, and fix t 0 and t 2. Then the intersection k t1 J i t t t is a onoial ideal of the for I for soe 0 d and d 2. Proof. If ij for soe j, then we have J ij j j 0, so k t1 J i t t t 0, and the desired equality follows fro Exaple 4.3 in this case. Thus, we assue for the reainder of the proof that ij for all j. Reorder the i t s if necessary to obtain the first equality in the next sequence, where 0 b 1 b 2 b d b d+1 k + 1: b k d i+1 1 d J it t t J i t t tb i t1 i1 J i i i i1 I The second step is fro Lea 4.6, and the third step is fro Lea 4.4.

12 DECOMPOSITIONS OF MONOMIAL IDEALS 4373 We deonstrate the algorith fro the proof of Lea 4.7 in the next exaple. Exaple 4.8. Let d 2. We show how to write the ideal I J J J J [ ] ] J J [J J in the for I. Define 2 0 and 2 2 by 1 ax , 1 1, 2 ax , and 2 0. Then we have I J J I. Leas 4.6 and 4.7 show how to siplify an arbitrary intersection of ideals of the for J i i i. The next leas are proved siilarly and show how to siplify an arbitrary su of these ideals. Lea 4.9. Let i b be given such that 1 i d and b 0. Given 1 b 0 and 1 b 2, there exist 0 and 2 such that b t1 J i t t J i. Specifically, we have { in 1 b if b 1 if b 0 Lea Let k be a positive integer. For t 1 k, let i t 1 d be given, and fix it 0 and t 2. Then k t1 J i t t t is a onoial ideal of the for J for soe 0 d and d 2. The next exaple is included to shed soe light on Lea Exaple Let d 2. We show how to write the ideal [ ] [ ] I J J J J J J J J in the for J. Define 2 0 and 2 2 by 1 in , 1 0, 2 in , and 2 0. Then we have I J J J. We are now in a position to prove the ain result of this section, which contains Theore 1.2 fro the introduction. Theore A onoial ideal I R has a finite -irreducible decoposition if and only if it can be expressed as a finite su of ideals of the for I. Proof. : Assue that I has a finite -irreducible decoposition. Theore 3.9 explains the first equality in the next sequence: I k J t t t1

13 4374 INGEBRETSON AND SATHER-WAGSTAFF k d J it t i t t it t1 i t 1 d d d k i 1 1 i 2 1 d i 1 1 d i 2 J it t i t t it i k 1 t1 d I i i i k 1 The reaining equalities are fro Fact 3.6, Lea 2.7(a), and Lea 4.7. Thus, the ideal I is a su of the desired for. : Assue that I is a finite su of ideals of the for I : I k I t t t1 k d J it t i t t it t1 i t 1 d i 1 1 i 2 1 d i 1 1 i 2 1 d d k J it t i t t it i k 1 t1 d d J i i i k 1 The second, third, and fourth steps in this sequence are fro Leas 4.4, 2.7(b), and 4.10, respectively. This expresses I as a finite intersection of ideals of the for J, so Theore 3.9 iplies that I has a finite -irreducible decoposition. The next exaple exhibits a onoial ideal in R that does not adit a finite -irreducible decoposition. The discussion of the case d 1 in the introduction shows every onoial ideal in this case is -irreducible. Thus, our exaple ust have d 2. The essential point for the exaple is that the graph of the line y 1 x that defines the exponent vectors of the generators of I is not a descending staircase which is the for required for an ideal to be a finite su of ideals of the for I. Note that any ideal defined by a siilar curve (e.g., any curve of the for y f x where f x is a non-negative, strictly decreasing, continuous function on the non-negative interval a b ) will have the sae property. Exaple Set d 2. We show that the ideal I X r Y 1 r 0 r 1 R does not adit a finite -irreducible decoposition. By Theore 4.12, it suffices to show that I cannot be written as a finite su of ideals of the for I. Suppose by way of contradiction that I k i1 I i i. If there are indices i and j such that I i i I j j, then we ay reove I j j fro the list of ideals without changing the su. Repeat this process for each pair of indices i, j such that I i i I j j to reduce to the case where no such containents occur in the su; since the list of ideals is finite, this process terinates in finitely any steps.

14 DECOMPOSITIONS OF MONOMIAL IDEALS 4375 We clai that for each real nuber r such that 0 <r<1, there is an index i such that I i i X r Y 1 r R I r 1 r 0 0. (This will iply that the index set 1 k cannot be finite, a contradiction.) The onoial X r Y 1 r is in I k i1 I i i by Fact 2.3(c). It follows that X r Y 1 r I i i for soe i. Since X r Y 1 r X r Y r I for all r r 0 such that r <r and r < 1 r, it follows readily that I i i X r Y 1 r R I r 1 r 0 0. Our final result shows that every onoial ideal of R has a possibly infinite -irreducible decoposition and can be written as a possibly infinite su of ideals of the for I. 2 Proposition there are equalities Let I be a onoial ideal with onoial generating set S. Then I I r 0 X r S where and J r 1 X r I Proof. The first equality is straightforward, using Exaple 4.3: I S R X r R I r 0 X r S X r S Fact 2.3(d) iplies that the ideal X r I J r 1 is a onoial ideal of R, so Fact 2.2(c) iplies that we need only show that I X I J r r 1. For the containent I X I J r r 1 let X I and X r I; we need to show that X J r 1, that is, that i >r i for soe i. Suppose by way of contradiction that i r i for all i. Then X r X R I, a contradiction. For the reverse containent I X I J r r 1, we show that R \ I R \ X I J r r 1. Let X R \ I. It follows that X is in the index set for the intersection X I J r r 1. Since i i for all i, we have X J 1,so ] X R \ [J 1 ( [[ ]]) [[ ]] R \ Jr 1 R \ Jr 1 X r I X r I as desired. ACKNOWLEDGMENTS We are grateful to the anonyous referee for her/his thoughtful coents. This aterial was supported by North Dakota EPSCoR and National Science Foundation Grant EPS Sean Sather-Wagstaff was supported in part by a grant fro the NSA. 2 Note that the case S is covered by the convention that the epty su of ideals is the zero ideal; the case R \ I is siilarly covered since the epty intersection contains the epty product which is the unit ideal.

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