Polynomials, -ideals, and the principal lattice Текст научной статьи по специальности «Математика»

УДК 512.54

Polynomials, a-ideals, and the Principal Lattice

Ali Molkhasi*

Institute of Mathematics and Mechanics, Academy of Sciences of Azerbaijan

F.Agaev, 9, Baku, 370141,

Azerbaijan Republic

Received 22.12.2010, received in revised form 11.02.2011, accepted 20.03.2011 Let R be a commutative ring with an identity, R be an almost distributive lattice and Ia (R) be the set of all a-ideals of R. If L(R) is the principal lattice of R, then R[Ia(R)] is Cohen-Macaulay. In particular, R[Ia (R)] [Xi , X2, • • • ] is WB-height-unmixed.

Keywords: Almost distributive lattice, principal lattice, a-ideals, multiplicative lattice; complete lattice, WB-height-unmixedness, Cohen-Macaulay rings, unmixedness.

Introduction

Lattices, in general, (specially multiplicative lattices), are natural abstractions of the set of ideals of a ring. However, Wihout a good notion of principal lattice, it is impossible to get very deep results, see [3]. Dilworth overcame this in [4], with a new notion of a principal element. Recall, that a multiplicative lattice is a complete lattice L with a commutative, associative multiplication which distributes over arbitrary joins and its largest element I, is the identity for the multiplication, see [4]. Basically, an element E of a multiplicative lattice L, is said to be meet-(join-)principal if (A A (B : E))E = (AE) A B (if (BE V A) : E = B V (A : E)) for all A and B in L. A principal element is an element that is both meet-principal and join-principal or AaE = (A : E)E and AE : E = AV(0 : E), for all A e L. A lattice L, is called a principal lattice, when each of its elements is principal. Here, the residual quotient of two elements A and B is denoted by A : B, so A : B = V{X e L|XB < A}. An almost distributive lattice (ADL), was introduced by U. M. Swamy and G. C. Rao in [11], as an algebra (R, V, A) of type (2,2), which satisfies almost all the properties of a distributive lattice, except possibly the commutativity of V, the commutativity of A and the right distributivity of V over A. W. H. Cornish studied in to [2], the properties of a-ideals in a distributive lattice, (see the next section for the definition). In this paper, the concept of an a-ideal in an ADL is introduced, analogous to the case of distributive lattices. In section 2, it is shown that, if R is a commutative ring with an identity and L(R) is the principal lattice, then R is a Cohen-Macaulay ring. In section 3, we prove that, if R is Cohen-Macaulay ring and if P is a distributive lattice, then R[P] is Cohen-Macaulay ring, where R[P] is the polynomial ring over P. Finally, in section 4, we conclude some properties of L(R).

1. The Principal Lattice and a-ideals

A Noether lattice is a modular multiplicative lattice satisfying the ascending chain condition in which every element is a join of elements called principal elements. The multiplication, meet,

* molkhasi@gmail.com © Siberian Federal University. All rights reserved

and join in a Noether lattice are supposed to mirror the multiplication, intersection, and sum of ideals. Because of this, a multiplicative lattice is defined to be a complete lattice L, containing a unite element I and a null element 0, and provided with a commutative, associative, join-distributive multiplication, for which I is an identity element. We will use A and V to denote meet and join, respectively, and < to denote lattice partial ordering, with < reserved for strict inequality.

Remark 1.1. Let R be a commutative ring with an identity, I and J be ideals of R, I + J and I J, be the ordinary sum and product of ideals. With these two operations as join and meet, the set of all ideals of a given ring, forms a complete modular lattice. Remember that, a principal lattice, is a lattice in which every element is principal. The following theorem is proved in [6].

Theorem 1.1. Let R be a commutative ring with identity. Then L(R) is a principal lattice, if and only if, R is a Noetherian multiplication ring.

Note that, a ring is called a multiplication ring, if every ideal of R is product of two ideals. Let M be a finitely generated module over a Noetherian ring R. We say that x G R is an M-regular element, if xg = 0 for g G M implies g = 0, in the other words, if x is not a zero-divisor on M. A sequence xi, • • • xr of elements of the ring R, is called an M-regular sequence or simply an M-sequence if the following conditions are satisfied:

(1) xi is an M/(x1, • • • , xi-1)M-regular element for i = 1,... ,r;

(2) M/(x1,... ,xr)M = 0.

Suppose I C R is an ideal with IM = M. The depth of I on M is maximal length of an M-regular sequence in I, denoted by depth(I,M). If R is a local ring with a unique maximal ideal m, we write depth(m), for depth(m, M).

Let R be a Noetherian local ring. A finitely generated R-module M, is a Cohen-Macaulay module, if depth(M) = dim(M). If R itself is a Cohen-Macaulay module, then it is called a Cohen-Macaulay ring. For the proof of the following theorem, see [9].

Theorem 1.2. Suppose R is a Noetherian multiplication ring. Then R is a Cohen-Macaulay ring.

2. Cohen-Macaulay and Unmixedness

We begin this section by a definition from Bourbaki.

Definition 2.1. A prime ideal P is an associated prime of I, if P = I : x for some x G R.

Remember that the height of a prime ideal P is the maximum length of the chains of prime ideals of the following form,

Pi C P2 C • • • C Pk = P.

We will denote the height of P by ht(P). An ideal I of R is said to be height-unmixed, if all the associated primes of I have equal height. That is ht(P) = ht(Q), for all P,Q G Ass(I), where Ass(I) denotes the set of associated primes of I. An ideal I is said to be unmixed if there are no embedded primes among the associated primes of I. That is, P C Q ^ P = Q, for all P,Q G Ass(I).1 We will say that an ideal is WB-height-unmixed, if it is height-unmixed with respect to the set of weak Bourbaki associated primes and an ideal is WB-unmixed if it is unmixed with respect to the set of weak Bourbaki associated primes. The set of weak Bourbaki

associated primes of an ideal I is denoted by Ass f (I). A prime ideal P is a weak Bourbaki associated prime of the ideal I if it is a minimal ideal of the form I: a, for some a G R.

Theorem 2.1. If R satisfies GPIT (generalized principal ideal theorem), then R is WB-height-unmixed if and only if R is WB-unmixed.

Proof. Suppose R is a ring which satisfies GPIT.

(^) Suppose R is WB-height-unmixed and let I be a height-generated ideal in R. In a ring which satisfies GPIT every ideal I satisfies ht(I) < l(I) where l(I) denotes the minimal number of generators of I. Thus, in a ring with GPIT, I is height-generated if and only if ht(I) = l(I) < <. To show that I is WB-unmixed, suppose P,Q g Assf (I), with P C Q. Since R is WB-height-unmixed so I is WB-height-unmixed. Thus, ht(P) = ht(Q) = ht(I) < <. So, P and Q are prime ideals with P C Q and ht(P) = ht(Q) < <. Thus, P = Q and so I is WB-unmixed. Therefore, R is WB-unmixed.

(^) Suppose R is WB-unmixed and let I be a height-generated ideal in R. As in the first part of this proof, we have ht(I) = l(I) < <. Let n = ht(I) = l(I). Note that, since R satisfies GPIT, we have ht(P) < n for all P g Min(I). However, since ht(I) < ht(P) for all P g Min(I) and ht(I) = n, we have ht(P) = n for all minimal associated prime P. Since I is WB-unmixed, we have Assf (I) is the set of all minimal associated primes of I. Therefore, ht(P) = n for all P G Assf (I) and thus, I is WB-height-unmixed. Therefore, R is WB-height-unmixed. □

Theorem 2.2. Let R be a Noetherian ring and let S = R[X1:X2,...], the ring of polynomials in the variables Xi,X2,.... For any prime ideal P in R we have ht(P) = ht(PS) where ht(PS) refers to the height of the ideal PS in S.

Proof. Note that the proof of this theorem depends only on the weaker condition that R is a strong S-ring ( see [8] for more information on strong S-rings). It is not necessary for the ring to be Noetherian. First, note that, we have trivially ht(P) ^ ht(PS), since the extensions of a chain of distinct prime ideals in R, is a chain of distinct prime ideals in S. For i ^ 1, let Ri = R[X1, X2,..., Xi]. So, S = limRj. Let Pi = PRi. Since R is Noetherian (and thus a strong S-ring), we have ht(P) = ht(Pi), see [8](Theorem 149, page 108). Now, suppose h(PS) > n, where n = ht(P). Then there is a chain of prime ideals

Qo C Qi C • • • C Qn+i = PS,

in S. For 1 < i < n +1, choose xi G Qi \ Qi-1. Since S = limRi, there is a positive integer j such that {x1, • • • , xn+1} G Rj. For 0 < i < n + 1, let Ti = Qi p| Ri. Then

To c T1 c • • • c Tn+1

is a chain of prime ideals in Rj. So, ht(Tn+1) > n +1. However, Tn+1 = Qn+1f) Rj = PSf)Rj = Pj and we have already noted that ht(Pj) = n, a contradiction. Therefore, ht(PS) = ht(P). □ In [1], it was shown that R[X1,X2,...] satisfies GPIT (if R is a Noetherian ring). The statement of this fact in [1] actually makes the assumption that R is a domain, however, the fact that R is a domain, is not necessary in the proof given in [1], so we will use the more general result. By applying 2.1 to this result, we get the following theorem

Theorem 2.3. Let R be a Cohen-Macaulay ring. Then R[X1,X2,...] is WB-height-unmixed.

Theorem 2.4. Let L be a Noetherian multiplicative lattice. Every element of L is principal element, if and only if, for all a ^ b, there is an element c G L, such that a = bc.

Proof. Suppose that elements of L are principal and let a,b G L and a ^ b. Then a = a n b = (a : b)b, and so c = (a : b). Conversely, it follows from (ACC), that each element of L is a join of a finite number of principal elements. Therefore, to prove the theorem it is sufficient to show that if m and n are principal elements of L, then m U n is principal. Let m be principal and let m ^ d, where d G L. Then m = cd, for some c G L, and since m is join principal ,(a U bcd) : cd = a : m U b, for all a,b G L. Hence ((a U bcd) : c) : d = a : m U b. However, (a U bd)c = ac U bdc ^ a U bcd, and so a U bd ^ (a U bcd) : c. Therefore, (a U bd) : d ^ a : m U b for all a,b G L. Thus, if m and n are principal elements of L, we have for all a,b G L,

(a U b(m U n)) : (m U n) ^ (a : m U b) n (a : n U b)

= ( a : m n a : n) U b = a : (m U n) U b.

Corollary 2.1. Let R be a commutative ring with an identity and L(R) be a Noether lattice. Every ideal of R is an principal element in L(R), if and only if, R is a multiplication ring.

For the proof of the following theorem, see [5].

Theorem 2.5. If R is Cohen-Macaulay ring, and if P is a distributive lattice, then R[P] is Cohen-Macaulay.

3. a-ideals and Cohen-Macaulay Rings

In this section we introduce the concept of an a-ideal in an ADL with zero, analogous to that in a distributive lattice [2]. An Almost Distributive Lattice (ADL) is an algebra (R, v, A) of type (1.2) satisfying:

1. (x V y) A z = (x A z) V (y A z)

2. x A (y V z) = (x A y) V (x A z)

3. (x V y) A y = y

4. (x V y) A x = x

5. x V (x A y) = x for any x,y, z G R

If R has an element 0, and satisfies 0 A x = 0 and 0 V x = 0 alogn with the above properties, then R is called an ADL with 0.

Definition 3.1. For any non-empty subset A of an ADL, R with 0, define A* = {x G R | a A x = 0, for all a G A}. Then A* is called the annihilator of A. For any a G R, we have {a}* = (a]*, where (a] is the principal ideal generated by a. For any 0 = A C R, we have clearly A n A* = (0].

For the proof of the next lemmas, see [10].

Lemma. For any non-empty subset A of R, A* is an ideal of R.

Lemma. For any non-empty subsets I, J of R, we have the following: 1■ If I C J, then J* C I * 2. I C I**

3 I ***_I *

4. (I v J)* = I* n J*.

Definition 3.2. Let R be a ADL with 0. An ideal I of R is called an a-ideal if (#]** C I for all x £ I.

We now denote the set of all a-ideal of an ADL R by Ia(R). If R is an ADL, then we know that (I(R), V, A) is a distributive lattice. But the set Ia(R) is not a sublattice of I(R).

Definition 3.3. A Noether lattice is said to be complete if it is complete in the topology of the Jacobson radical.

In [7], the following theorem is proved.

Theorem 3.1. Let (L, m) be a distributive local Noether lattice of dimension d. Then L is complete in the m-adic topology.

Let R be a local Noetherian ring with the maximal ideal M. Then L(R), the lattice of ideals of R, is a local Noether lattice and also L(R) is a complete modular lattice. A ring R is called an arithmetical ring, if L(R) is distributive.

Corollary 3.1. If (R, m) is a local Noetherian ring and is arithmetical ring, then L(R) is a complete in the m-adic topology.

Proof. This is immediate from Remark 1.1 and Theorem 3.1 and if L is an ADL, then L(R) is a distributive lattice. □

Corollary 3.2. If (R, m) is local Noether lattice and is an ADL, then I(R) is a complete in the m-adic topology.

In [10], it is proved that, if R is an ADL with 0, then Ia(R) forms a distributive lattice. So we have

Theorem 3.2. Let R be a commutative ring with an identity and let Ia(R) be the set of all a-ideal of an ADL R. If L(R) is a principal lattice, then R[Ia(R)][X1, X2,...] is WB-height-unmixed.

Proof. Since L(R) is a principal lattice, R is Cohen-Macaulay . By assumption, Ia(R) is a distributive lattice. Thus the R[Ia(R)] is Cohen-Macaulay and R[Ia(R)][X1, X2,...] is WB-height-unmixed. □

References

[1] D.F.Anderson, D.E.Dobbs, Coherent Mori domains and the principal ideal theorem, Communications in Algebra, 15(1987), 1119-1156.

[2] W.H.Cornish, Annulets and a-ideals in distributive lattices, J. Austral. Math. Soc., 15(1973), 70-77.

[3] R.P.Dilworth, Dilworth's early papers on residuated and multiplicative lattices. The Dil-worth theorem, Birkhauser, Boston, 1990, 387-390.

[4] R.P.Dilworth, Abstract commutative ideal theory, Pacific J. Math., 12(1962), 481-498.

[5] C.de Concini, D.Eisenbud, D.Procesi, Hodge algebras, Asterisque, 91(1982).

[6] M.F.Janowitz, Principal mutiplicative lattices, Pacific J. Math., 33(1970), 653-656.

[7] E.W.Johnson, J.Johnson, Representations of complete regular local Noetherian lattices, Tamkang journal of mathematics, 39(2008), no. 2, 137-141.

[8] I.Kaplansky, Commutative rings, Alyn and Bacon, 1970.

[9] R.Naghipour, H.Zakrei, N.Zamani, Cohen-Macaulayness of multiplication rings and modules, Colloquium Mathematicum, 95(2002), 133-138.

[10] G.C.Rao, a-ideals in almost distributive lattices, Int. J. Contemp. Math. Sciences, 4(2009), 457-466.

[11] U.M.Swamy, G.C.Rao, Almost distributive lattices, J. Austral. Math. Soc., Ser. A, 31(1981), 77-91.

Полиномы, а-идеалы и главные решетки

Али Молхаси

Пусть R — коммутативное кольцо с единицей, R — почти дистрибутивная решетка и Ia (R) — множество всех а-идеалов в R. Если L(R) — главная решетка R, то R[Ia(R)] — кольцо Коэна-Маколея. В частности, R[Ia(R)][X1,X2, • • • ] — WB-высота несмешанности.

Ключевые слова: почти дистрибутивные решетки, главные решетки, а-идеалы, WB-высота несмешанности, полные решетки, Коэна-Маколея кольца, несмешанность.