# Residuation Properties and Weakly Primary Elements in Lattice Modules

Authors: C. S. Manjarekar; U. N. Kandale
Journal: Algebra
 (2014)

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## Abstract

We obtain some elementary residuation properties in lattice modules and obtain a relation between a weakly primary element in a lattice moduleM and weakly prime element of a multiplicative lattice L.


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## Body

## 1. Introduction

A multiplicative latticeL is a complete lattice provided with commutative, associative, and join distributive multiplication in which the largest element I acts as a multiplicative identity. An element a∈L is called proper if a<1. A proper element p of L is said to be prime if ab≤p implies a≤p or b≤p. If a∈L, b∈L, (a:b) is the join of all elements c in L such that cb≤a. A proper element p of L is said to be primary if ab≤p implies a≤p or bn≤p for some positive integer n. If a∈L, the radical of a is denoted by a=∨{x∈L∣xn⩽a,n∈Z+}. An element a∈L is called compact if a⩽∨αbα implies a⩽bα1∨bα2∨⋯∨bαn for some finite subset {α1,α2,…,αn}. Throughout this paper, L denotes a compactly generated multiplicative lattice with 1 compact and every finite product of compact elements is compact. We will denote by L* the set compact elements of L. A nonempty subset F of L* is called a filter of L* if the following conditions are satisfied:(1)
x,y∈F implies xy∈F;(2)
x∈F, x⩽y implies y∈F.LetF(L*) denote the set of all filters of L. For a nonempty subset {Fα}⊆F(L*), define ⋓Fα={x∈L*∣x≥f1f2⋯fn∈Fαi, for some i=1,2,…,n}. Then it is observed that F(L*)=〈F(L*),⋓,∩〉 is a complete distributive lattice with ⋓ as the supremum and the set theoretic ⋂‍ as the infimum. For a∈L* the smallest filter containing a is denoted by [a) and it is given by [a)={x∈L*∣x≥an for some nonnegative integer n}. For a filter F∈F(L*) we denote 0F=∨{x∈L*∣xs=0,fors∈F}.LetM be a complete lattice and L a multiplicative lattice. Then M is called L-module or module over L if there is a multiplication between elements of L and M written as aB where a∈L and B∈M which satisfies the following properties:(1)
∨αaαA=∨αaαA,  ∀aα∈L, A∈M;(2)
a∨αAα=∨αaAα,  ∀a∈L, Aα∈M;(3)
abA=abA, ∀a,b∈L, A∈M;(4)
IB=B;(5)
OB=OM, for all a,aα,b∈L and A,Aα∈M, where I is the supremum of L and O is the infimum of L. We denote by 0M and IM the least element and the greatest element of M. Elements of L will generally be denoted by a,b,c,… and elements of M will generally be denoted by A,B,C,….LetM be an L-module. If N∈M and a∈L then (N:a)=∨{X∈M∣aX⩽N}. If A,B∈M, then (A:B)=∨{x∈L∣xB⩽A}. An L-module M is called a multiplication L-module if for every element N∈M there exists an element a∈L such that N=aIM. In this paper a lattice module M will be a multiplication lattice module. A proper element N of M is said to be prime if aX⩽N implies X⩽N or aIM⩽N; that is, a⩽(N:IM) for every a∈L, X∈M. If N is a prime element of M then (N:IM) is prime element of L [1]. An element N<IM in M is said to be primary if aX⩽N implies X⩽N or anIM⩽N; that is, an⩽(N:IM) for some integer n. An element N of M is called a radical element if (N:IM)=(N:IM).


## 2. Residuation Properties

We prove some elementary properties of residuation. Such results are obtained by Anderson and others [2] for compactly generated multiplicative lattices.Theorem 1.
Let L be a multiplicative lattice andM a multiplication lattice module over L. For x,y∈L and Z,A,B∈M, where (0M:IM) is a radical element, one has the following identities:(i)
x⩽y implies (0M:y)⩽(0M:x) and 0M:(0M:x)⩽0M:(0M:y);(ii)
x⩽0M:(0M:x);(iii)
0M:[0M:(0M:x)]=(0M:x);(iv)
(0M:x)=(0m:xn) for every n∈Z+;(v)
0M:(0M:x)∧0M:(0M:y)=0M:(0M:xy)=0M:[0M:(x∧y)];(vi)
(0M:a)=0M implies (0M:an)=0 for every n∈Z+;(vii)
x∨y=1 implies (0M:x)∨(0M:y)=0M:(x∧y)=0M:xy;(viii)
ForZ in M, Z⩽0M:(0M:Z);(ix)
A⩽B implies (0M:B)⩽(0M:A);(x)
0M:[0M:(0M:A)]=0M:A;(xi)
0M:xIM=0M:xnIM for some positive integer n.Proof.
(i)  We have(0M:y)=∨{A∈M∣yA=0M}. As x⩽y, xA⩽yA, so yA=0M implies xA=0M. This shows that (0M:y)⩽(0M:x). Next part 0M:(0M:x)⩽0M:(0M:y) is obvious.
(ii)  We havex(0M:x)=x{∨{A∣xA=0M}}=∨{xA∣xA=OM}=0M. This implies that x⩽0M:(0M:x).
(iii)  We have, by (ii),x⩽0M:(0M:x). So by (i), 0M:[0M:(0M:x)]⩽(0M:x). If A∈M, A(0M:A)=∨{xA∣xA=0M}=0M. So, A⩽0M:(0M:A). Hence, (0M:x)∈M implies (0M:x)⩽0M:[0M:(0M:x)]. Therefore, 0M:[0M:(0M:x)]=(0M:x).
(iv)  We have(0M:x)=∨{A∈M∣xA⩽0M}. As xn⩽x, we have (0M:x)⩽(0M:xn) for every n∈Z+. Let Z=(0M:xn) where Z∈M. As M is a multiplication lattice module Z=aIM for some a∈L. So xnanIM⩽xnaIM=0M. Hence xa⩽(0M:IM)=(0M:IM). So xaIM⩽0M; that is, Z⩽(0M:x), and hence (0M:x)=(0M:xn) for each positive integer n.
(v)  Note thatxy⩽x,y. Therefore by (i), 0M:(0M:xy)⩽0M:(0M:x)∧0M:(0M:y). Let z=0M:(0M:x)∧0M:(0M:y). Now we have z⩽0M:(0M:x) and z⩽0M:(0M:y). Then, z(0M:x)=0M and z(0M:y)=0M. This implies that (0M:x)⩽(0M:z), (0M:y)⩽(0M:z). Therefore, we have (0M:xy)=(0M:x):y⩽(0M:z):y=(0M:yz) [3]. Hence, (0M:xy)⩽(0M:y):z⩽(0M:z):z=(0M:z2)=(0M:z), by (iv). That is, z(0M:xy)=0M or z⩽0M:(0M:xy). Consequently, 0M:(0M:xy)=0M:(0M:x)∧0M:(0M:y). Note that x∧y⩽x,y. Then by (i), we have 0M:(0M:(x∧y))⩽0M:(0M:x)∧0M:(0M:y). Let z=0M:(0M:x)∧0M:(0M:y). We have 0M:(x∧y)⩽0M:xy⩽0M:z2=0M:z, by (iv). This shows that z[0M:(x∧y)]⩽0M. Thus, z⩽0M:[0M:(x∧y)]. Consquently, 0M:[0M:(x∧y)]=0M:(0M:x)∧0M:(0M:y).
(vi)  Assume that(0M:a)=0M. We have, by (iv), (0M:a)=(0M:an) for every n∈Z+. But (0M:a)=0M. Thus, (0M:an)=0M for every n∈Z+.
(vii)  Assume thatx∨y=1. As x∧y⩽x,y, we have (0M:x)⩽(0M:(x∧y)), (0M:y)⩽(0M:(x∧y)). Therefore, (0M:x)∨(0M:y)⩽0M:(x∧y). Let Z in M be such that Z⩽0M:(x∧y). Then, Zxy⩽Z(x∧y)⩽0M. This gives Zx⩽(0M:y) and Zy⩽(0M:x). Then we have, for Z in M, Z=Z.1=Z(x∨y)=Zx∨Zy⩽(0M:x)∨(0M:y). Consequently, 0M:(x∧y)=(0M:x)∨(0M:y) For the next part, xy⩽x,y where x,y∈L. We have (0M:x)⩽(0M:xy), (0M:y)⩽(0M:xy). Therefore, we get (0M:x)∨(0M:y)⩽(0M:xy). Let Z be an element of M such that Z⩽0M:xy. Then, xyZ⩽0M. This gives xZ⩽(0M:y) and yZ⩽(0M:x). Then we have Z=1.Z=(x∨y)Z=xZ∨yZ⩽(0M:x)∨(0M:y). Consequently, 0M:(xy)=(0M:x)∨(0M:y).
(viii)  We have, forZ in M,  Z(0M:Z)=Z[∨{a∈L∣aZ=0M}]=∨{aZ∣a∈L and aZ=0M}. This implies that Z(0M:Z)=0M. This shows that, for Z in M, Z⩽0M:(0M:Z).
(ix)  LetA⩽B. We have x⩽(0M:B) implies xB=0M. A⩽B implies xA⩽xB=0M. So, x⩽(0M:A). Therefore, (0M:B)⩽(0M:A).
(x)  As(0M:A)∈L, by (ii), we get (0M:A)⩽0M:[0M:(0M:A)]. But A⩽0M:(0M:A). Therefore, (0M:A)≥0M:[0M:(0M:A)]. Thus, 0M:[0M:(0M:A)]=(0M:A).
(xi)  Asxn⩽x, we have xnIM⩽xIM. So (0M:xIM)⩽(0M:xnIM). Let z=(0M:xnIM) where z∈L. Then zxnIM⩽0M; that is, xnzIM=0M. Therefore, xnznIM=(xz)nIM⩽xnzIM=0M. Hence, xz⩽(0M:IM)=(0M:IM), since 0M is a radical element. This implies that z⩽(0M:xIM). Consequently, (0M:xnIM)⩽(0M:xIM). Hence, (0M:xIM)=(0M:xnIM).LetM* denote the set of compact elements of M. We define 0FM=∨{X∈M*∣sX=0M for some s∈F}. Such concept is defined for multiplicative lattices by Anderson and others [2].We obtain the necessary and sufficient condition for an element of a lattice moduleM to be contained in 0FM.Theorem 2.
LetF⊆L be a filter of F(L*) and let X be a compact element of M. Then X⩽0FM if and only if sX=0M for some s∈F.Proof.
AsX is compact element and X⩽0FM, there exist finite number of elements X1,X2,…,Xn such that X⩽X1∨X2∨⋯∨Xn where siXi=0M for some si∈F. Let s=s1s2⋯sn. Then by hypothesis, finite product of compact elements is compact, and hence, s is a compact element. As F is a filter, s∈F. Also, s⩽si for each i. Hence, we have sX⩽s(X1∨X2∨⋯∨Xn)=sX1∨⋯∨sXn⩽s1X1∨s2X2∨⋯∨snXn=0M. Conversely, suppose sX=0M for some s∈F. Hence, X⩽0FM obviously.Now we obtain an equivalent formulation for0FM.Theorem 3.
ForF∈F(L*), 0FM=∨{(0M:x)∣x∈F}.Proof.
LetA=∨{(0M:x)∣x∈F}. Let B be a compact element of M such that B⩽0FM. Then by Theorem 2 there exists x∈F such that xB=0M. So, B⩽(0M:x). Hence, B⩽A=∨{(0M:x)∣x∈F}. Therefore, 0FM⩽∨{(0M:x)∣x∈F}=A. Let B be a compact element such that B⩽A. Then, there exist x1,x2,…,xn∈F such that B⩽0M:x1∨0M:x2∨⋯∨(0M:xn). By compactness of B, there exist compact elements Y1,Y2,…,Yn∈M such that xiYi=0M and B⩽Y1∨Y2∨⋯∨Yn. As F is a filter, we have x=x1x2⋯xn∈F. Hence, xB⩽xY1∨Y2∨⋯∨Yn⩽x1Y1∨x2Y2⋯∨xnYn=0M. Thus, B⩽0FM. This shows that 0FM=∨{(0M:x)∣x∈F}.Theorem 4.
ForF1,F2∈F(L*)(1)
F1⊆F2 implies 0F1M⩽0F2M;(2)
0F1M∧0F2M=0(F1⋂‍F2)M.Proof.
(1)  LetF1⊆F2 and let X be a compact element of M such that X⩽0F1M. Then by Theorem 2  sX=0M, for some s∈F1. By assumption, s∈F2, sX=0M. Hence, X⩽0F2M and 0F1M⩽0F2M.
(2)  We haveF1⋂‍F2⊆F1,F2. Hence by 1, 0(F1⋂‍F2)M⩽0F1M,0F2M. Thus, 0(F1⋂‍F2)M⩽0F1M∧0F2M. Let X be a compact element in M such that X⩽0F1M∧0F2M. Then X⩽0F1M and X⩽0F2M. By Theorem 2, there exist elements s∈F1 and r∈F2 such that sX=0M and rX=0M. So X⩽(0M:s) and X⩽(0M:r). Consequently, X⩽(0M:s)∧(0M:r)=0M:(s∨r) [3]. As r and s are compact elements, s∨r is also compact. As F1 and F2 are filters, s∨r∈F1 and s∨r∈F2. This implies that s∨r∈F1⋂‍F2. Thus, X⩽0M:(s∨r) where s∨r∈F1⋂‍F2. That is, (s∨r)X=0M. Hence, by Theorem 2, X⩽0(F1⋂‍F2)M. That is, 0F1M∧0F2M⩽0(F1⋂‍F2)M. Thus, 0F1M∧0F2M=0(F1⋂‍F2)M.


## 3. Weakly Prime and Weakly Primary Elements

We introduce the concepts of weakly prime and weakly primary elements in lattice modules. These concepts are generalisations of weakly prime elements in multiplicative lattices [4] and weakly primary ideals [5].Definition 5.
A proper elementN of M is said to be weakly prime element if 0≠aX⩽N implies X⩽N or aIM⩽N for every a∈L, X∈M.Definition 6.
A proper elementN of M is said to be weakly primary element if 0≠aX⩽N implies X⩽N or anIM⩽N for every a∈L, X∈M.Example 7.
LetM denote a cyclic Z-module Z/6Z and let L=L(Z) denote the set of all ideals of Z and let L(Z/6Z) denote the set of all submodules of Z/6Z. Then L(Z/6Z) is a lattice module over L. Let N={0}. Then N is a weakly prime element of M.Example 8.
LetR=Z denote the set of integers and let L(R) denote the set of all ideals of R. Then L(R) is a multiplicative lattice. Let M=Z×Z. Then M is a module over Z. Let L(M) denote the set of all submodules of M. Then L(M) is a lattice module over L(R). Let L=(4,0)Z+(0,1)Z; then L is a weakly primary element of L(M); however it is not weakly prime element of L(M), because 0≠(2)(2,1)⊆L. But neither (2)M⊆L nor (2,1)∈L; see [6].The next result gives the relation between weakly primary element ofM and weakly primary element of L.The study of such concepts for submodules is carried out by Eid [6].Theorem 9.
LetL be a multiplicative lattice, M a lattice module over L, and N a weakly primary element of M. Then (N:IM) is a weakly primary element of L.Proof.
SupposeN is a weakly primary element of M. Let 0≠ab⩽(N:IM) and aN:IM where a,b∈L. This implies that aIMN. But N is a weakly primary element and 0≠abIM⩽N and aIMN. So bnIM⩽N for some n∈Z+; that is, bn⩽(N:IM). Hence, (N:IM) is a weakly primary element of L.Now we obtain the relation between weakly primary element ofM and weakly prime element of L where L is assumed to be without divisors of zero.Theorem 10.
LetL be a multiplicative lattice and M a lattice module over L and suppose N is a weakly primary element of M. Then (N:IM) is a weakly prime element of L.Proof.
SupposeN is a weakly primary element of M. Then by Theorem 9  (N:IM) is a weakly primary element of L. Let 0≠ab⩽(N:IM) with a(N:IM). Hence an(N:IM) for any n∈Z+. That is, anIMN for any n∈Z+. We have 0≠(ab)k⩽(N:IM) for some k∈Z+. That is, 0≠akbkIM⩽N where akIMN and N is a weakly primary element M. Hence, (bk)lIM⩽N for some l∈Z+. That is, bmIM⩽N where kl=m∈Z+. Thus (N:IM) is a weakly prime element of L.



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