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IJMMS 26:3 (2001) 161–166

PII. S0161171201004604

http://ijmms.hindawi.com

© Hindawi Publishing Corp.

LATTICE MODULES HAVING SMALL COFINITE IRREDUCIBLES

E. W. JOHNSON, JOHNNY A. JOHNSON, and MONTY B. TAYLOR

(Received 16 February 2000)

Abstract. We introduce the concept of small cofinite irreducibles in Noetherian lattice

modules and obtain several characterizations of this property.

2000 Mathematics Subject Classification. 06F10, 06F05.

Let L be a multiplicative lattice and let M be an L-module with greatest element M.

Recallfrom[5]thatforanelementaofL,Rad(a) =?{x ∈ L | xn≤ a for some positive

integer n}. For an element B of M, we define Rad(B) to be Rad(B : M), that is, Rad(B) =

?{x ∈ L | xnM ≤ B for some positive integer n}. An element Q of M is defined to be

primary if for all b ∈ L and C ∈ M, bC ≤ Q implies either b ≤ Rad(Q) or C ≤ Q.

Lemma 1. Let L be an r-lattice, let M be an L-module with greatest element M, and

let A and B be elements of M. Then Rad(A∧B) = Rad(A)∧Rad(B).

Proof. We have

Rad(A∧B) = Rad?(A∧B) : M?= Rad(A : M∧B : M)

= Rad(A : M)∧Rad(B : M) = Rad(A)∧Rad(B),

(1)

where the third equality follows from [5, Lemma 2.2].

Lemma 2. Let L be a totally quasi-local lattice with maximal element m, let M be an

L-module, and let Q be an element of M. If Rad(Q) = m, then Q is primary.

Proof. Suppose Rad(Q) = m. Also suppose that b ∈ L and C ∈ M such that bC ≤

Q and C ? Q. Then b ≠ I, and since L is totally quasi-local, it follows that b ≤ m =

Rad(Q). Thus, Q is primary.

Let L be a totally quasi-local lattice with maximal element m and let M be an L-

module. For an element Q of M, Q is said to be m-primary if Rad(Q) = m.

Lemma 3. Let L be a local Noether lattice with maximal element m, let M be an L-

module with greatest element M, and let Q be an element of M different from M. Then

Q is m-primary if and only if there exists a positive integer n such that mnM ≤ Q.

Proof. Suppose Q is m-primary. Since Rad(Q : M) = m, there exists a positive

integer n such that mn≤ Q : M. Thus, mnM ≤ Q. Conversely, suppose that there

exists a positive integer n such that mnM ≤ Q. Then m = Rad(mnM) ≤ Rad(Q) ≤ m,

and so Q is m-primary.

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E. W. JOHNSON ET AL.

Let L be a local Noether lattice with maximal element m and let M be a Noetherian L-

module with greatest element M. Define a metric d (the m-adic metric) on M as follows:

d(A,B) = 0 if A∨mnM = B ∨mnM for all nonnegative integers n, and otherwise,

d(A,B) = 2−s(A,B), where s(A,B) = sup{n | A∨mnM = B ∨mnM}. This metric gives

rise to the m-adic completions of L and M (see [4]).

Lemma 4. Let L be a local Noether lattice with maximal element m, let M be a

Noetherian L-module with greatest element M, and let Q be an element of M. Then Q

is m-primary if and only if M/Q is finite dimensional.

Proof.

integer n such that mnM ≤ Q. Since M/mnM is finite dimensional [1, Corollary 5.2],

if follows that M/Q is finite dimensional.

On the other hand, suppose that Q is not m-primary. Then by Lemma 3, mnM ? Q

for all positive integers n. It follows that {Q∨mnM} is a strictly decreasing sequence

of elements of M with meet Q, so M/Q is not finite dimensional.

Suppose that Q is m-primary. Then by Lemma 3, there exists a positive

Let L be a local Noether lattice with maximal element m, let M be a Noetherian

L-module with greatest element M. We say that M has small cofinite irreducibles if

for every positive integer n, there exists a meet-irreducible element Q of M such that

Q ≤ mnM and M/Q is finite dimensional.

Theorem 5. Let L be a local Noether lattice with maximal element m and let M be

a Noetherian L-module with greatest element M. Then the following are equivalent:

(1) M has small cofinite irreducibles.

(2) For every positive integer n, there exists a meet-irreducible m-primary element

Q of M such that Q ≤ mnM.

(3) For every m-primary element Q?of M, there exists a meet irreducible m-primary

element Q of M such that Q ≤ Q?.

(4) 0 is a closure point in the set of all meet-irreducible m-primary elements of M in

the m-adic topology on M.

Proof.

ducibles. Suppose also that n is a positive integer. Then there exists a meet-irreducible

element Q of M such that Q ≤ mnM and M/Q is finite dimensional. By Lemma 4, we

have that Q is m-primary, so (2) holds.

We next show that (2) implies (4). Suppose that (2) holds and that ? > 0. Choose n

to be a positive integer satisfying 2−n< ?. Using (2), there exists a meet-irreducible

m-primary element Q of M such that Q ≤ mnM. Thus Q ∨ mnM = mnM, and so

d(Q,0) ≤ 2−n< ?. Therefore, 0 is a closure point in the set of meet-irreducible m-

primary elements of M in the m-adic topology on M.

Now we show that (4) implies (3). Suppose that (4) holds and that Q?is an m-primary

element of M. By Lemma 3, there exists a positive integer n such that mnM ≤ Q?. Since

0 is a closure point in the set of meet-irreducible m-primary elements of M in the m-

adic topology on M, there exists a meet-irreducible m-primary element Q of M such

that d(Q,0) ≤ 2−n. Hence, Q∨mnM = mnM, and so it follows that Q ≤ mnM. Thus

the meet-irreducible m-primary element Q satisfies Q ≤ Q?.

We begin by showing that (1) implies (2). Suppose M has small cofinite irre-

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Finally, we show that (3) implies (1). Suppose that (3) holds and that n is a pos-

itive integer. Since mnM is m-primary, then by (3), there exists a meet-irreducible

m-primary element Q of M such that Q ≤ mnM. Also, by Lemma 4, M/Q is finite

dimensional. Thus, M has small cofinite irreducibles.

Theorem 6. Let L be a local Noether lattice with maximal element m and let M be a

Noetherian L-module with greatest element M. Then M has small cofinite irreducibles

if and only if there exists a decreasing sequence {Qn} of meet-irreducible m-primary

elements of M such that for each m-primary element Q?of M, there exists a positive

integer n such that Qn≤ Q?.

Proof.Suppose that m has small cofinite irreducibles. Since mM is an m-primary

element of M, use (2) to pick Q1to be a meet-irreducible m-primary element of M such

that Q1≤ mM. For n > 1, recursively define Qnas follows: choose Qnto be a meet-

irreducible m-primary element of M such that Qn≤ Qn−1∧mnM, which is possible

by Lemma 1 since

Rad?Qn−1∧mnM?= Rad?Qn−1

?∧Rad?mnM?= m,

(2)

and so Qn−1∧mnM is an m-primary element of ˙ M. By construction, {Qn} is a de-

creasing sequence of meet-irreducible m-primary elements of M. Moreover, if Q?is

an m-primary element of M, then by Lemma 3 there exists a positive integer n such

that mnM ≤ Q?, and so Qn≤ Q?.

Conversely, suppose that there exists a decreasing sequence {Qn} of meet-

irreducible m-primary elements of M such that for all m-primary elements Q?of

M, there exists a positive integer n such that Qn≤ Q?. We immediately have that (2)

holds since for each positive integer n, mnM is an m-primary element of M. Thus, by

Theorem 5, M has small cofinite irreducibles, which completes the proof.

Let L be a local Noether lattice with maximal element m and let M be a Noetherian

L-module with greatest element M. Following [2], L∗denotes the set of all formal sums

?∞

i=1aiof elements of L such that

ai= ai+1∨mi

(3)

for all positive integers i. On L∗, define

∞

?

i=1

ai≤

∞

?

i=1

bi

if and only if ai≤ bi∀i,

∞

?

i=1

ai

∞

?

i=1

bi

=

∞

?

i=1

?aibi∨mi?.

(4)

For an element a of L, a∗denotes the element

a local Noether lattice with maximal element m∗=?∞

is a collection of representatives of equivalence classes of Cauchy sequences of L

with the m-adic metric and in fact is the completion of L with this metric. Additional

properties can be found in [2]. Similarly, M∗denotes the set of all formal sums?∞

?∞

i=1(a ∨ mi) of L∗. Then L∗is

i=1m. It can be seen that L∗

i=1Bi

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E. W. JOHNSON ET AL.

of elements of M such that

Bi= Bi+1∨miM

(5)

for all positive integers i. On M∗, define

∞

?

i=1

Bi≤

∞

?

i=1

Ci

if and only if Bi≤ Ci∀i,

∞

?

i=1

ai

∞

?

i=1

Bi

=

∞

?

i=1

aiBi∨miM.

(6)

It is known [1] that M∗is a Noetherian L∗-module with greatest element M∗=?∞

For an element B of M, B∗denotes the element?∞

is an element of M∗, then C(B) denotes the element?∞

Theorem 7. Let L be a local Noether lattice with maximal element m and let M be a

Noetherian L-module with greatest element M. Then the L-module M has small cofinite

irreducibles if and only if the L∗-module M∗has small cofinite irreducibles.

i=1M.

i=1Bi

i=1B∨miM of M∗. Also, if B =?∞

i=1Biof M.

Proof.

integer n, mnM contains an irreducible m-primary element Q, then choose i so that

miM ≤ Q. Then the element of M∗/(m∗)iM∗corresponding to Q is irreducible and

m∗-primary. The argument is reversible.

For any positive integer i, M/miM ? M∗/(m∗)iM∗. If for every positive

Let R be a local Noetherian ring with maximal ideal m and let M be a Noetherian

R-module. Then the R-module M is said to have small cofinite irreducibles if for every

positive integer n, there exists an irreducible submodule Q of M such that Q ⊆ mnM

and M/Q has finite length. Let L(R) denote the lattice of ideals of R and let L(M)

denote the lattice of R-submodules of M. Since the set of irreducible submodules of

M is precisely the set of meet-irreducible elements of the L(R)-submodule L(M), we

immediately have the following theorem.

Theorem 8. Let R be a local Noetherian ring with maximal element m and let M

be a Noetherian R-module. Then the R-module M has small cofinite irreducibles if and

only if the L(R)-module L(M) has small cofinite irreducibles.

For a Noetherian module M over a local ring R with maximal ideal m, we let M∗

and R∗denote the completions of M and R, respectively, in the m-adic topology.

Theorem 9. Let R be a local Noetherian ring with maximal element m and let M be

a Noetherian R-module. Then the following statements are equivalent:

(i) The R-module M has small cofinite irreducibles.

(ii) The R∗-module M∗has small cofinite irreducibles.

(iii) The L(R)-module L(M) has small cofinite irreducibles.

(iv) The L(R∗)-module L(M∗) has small cofinite irreducibles.

(v) The L(R)∗-module L(M)∗has small cofinite irreducibles.

Proof.

alence of (ii) and (iv). The equivalence of (iii) and (v) follows from Theorem 7. The

equivalence of (iv) and (v) is established in [3].

The equivalence of (i) and (iii) follows from Theorem 8. So does the equiv-

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In Theorem 8, we showed that the lattice of submodules of a module having small

cofinite irreducibles is a lattice module having small cofinite irreducibles. We conclude

this paper by giving an example of a module having small cofinite irreducibles which

is not the lattice of submodules of any module.

Let L be the local Noether lattice with maximal element m in which the quotient

m/m2has exactly two points, e and h. Further, assume each quotient mn/mn+1has

exactly two points for each n, with eihj= erhsif i+j = r +s and j and s are both

even or j and s are both odd.

I

m

e

h

m2

e2= h2

eh

m3

e3= eh2

h3= e2h

m4

e4= e2h2= h4

eh3= e3h

m5

0

(7)

It is clear that every power of m contains an irreducible m-primary element. If L is the

lattice of submodules of any module, then every cyclic submodule ≠ m, I is contained

in e or h. Then m = e∪h is a submodule, with e ? h and h ? e, which is impossible.