Baer Elements in Lattice Modules

Vol. 8, No. 3, 2015, 332-342

ISSN 1307-5543 – www.ejpam.com

Baer Elements In Lattice Modules

C S Manjarekar

_{, U N Kandale}

1_{Department of Mathematics, Shivaji University, Kolhapur, India}

2_{Department of General Engineering, Sharad Institute, Shivaji University, Kolhapur, India}

Abstract. Let L be a compactly generated multiplicative lattice with 1 compact in which every finite product of compact elements is compact and M be a module over L. In this paper we generalize the concepts of Baer elements,∗-elements and closed elements and obtain the relation between∗-elements and Baer elements and also closed elements and Baer elements. Some characterization are also obtain for closed elements of M and minimal prime elements of M.

2010 Mathematics Subject Classifications: 13A99

Key Words and Phrases: Prime element,primary element,lattice modules,Baer element, ∗-element, closed element.

1. Introduction

A multiplicative lattice L is a complete lattice provided with commutative, associative and join distributive multiplication in which the largest element 1 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 a b ≤ p implies a ≤ p or b ≤ p. Ifa ∈ L, b ∈ L, (a : b) is the join of all elements c in L such thatc b ≤ a. A proper element p of L is said to be primary ifa b≤ p impliesa ≤ por bn ≤ p for some positive integer n. If a ∈ L thenpa = ∨{x ∈ L | xn ¶ a,n ∈ Z₊}. An element a∈ L is called a radical element if a= pa. An element a ∈ L is called compact if a¶_∨

αbα impliesa¶ bα1∨bα2∨. . .∨bαnfor 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 shall denote byL_∗the set compact elements of L. A nonempty subset F ofL_∗is called a filter ofL_∗if the following conditions are satisfied,

(i) x,y∈F impl ies x y∈F (ii) x _∈F,x ¶ y impl ies y_∈F. ∗_{Corresponding author.}

Email addresses:[email protected](C Manjarekar),[email protected](U Kandale)

Let F(L_∗) denote the set of all filters of L. For a nonempty subset {Fα} ⊆ F(L_∗), define

⋒F_α = _{x _∈ L_{∗ |} x _≥ f₁f_{2· · ·}f_{n ∈} F_α

i, for somei = 1, 2, . . . ,n}. Then it is observed that,

F(L_∗) =_〈F(L_∗),⋒_{,∩〉is a complete distributive lattice with} ⋒_{as the supremum and the set}

theroreticTas the infimum. Fora∈L_∗the smallest filter containing a is denoted by[a) and it is given by[a) =_{x _∈L_{∗ |} x _≥ anfor some nonnegative integern_}. For a filter F _∈F(L_∗) we denote,0F=∨{x ∈L∗|xs=0, fors∈F}.

Let M be a complete lattice and L be 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 aBwhere a∈LandB∈M which satisfies the following properties,

(i) (_∨

αaα)A=∨αaαA ∀aα∈L, A∈M (ii) a(∨

αAα) =∨αaAα ∀a∈L, Aα∈M (iii) (a b)A=a(bA) _∀a,b∈L, A∈M (iv) 1B=B

(v) 0B=0_M for alla,a_α,b∈L andA,A_α ∈M, where 1 is the supremum of L and 0 is the infimum of L. We denote by 0MandIMthe least element and the greatest element of M. Elements of L will generally be denoted bya,b,c, . . . and elements of M will generally be denoted byA,B,C. . ..

Let M be a L-module. IfN∈M anda∈Lthen(N:a) =∨{X ∈M|aX ¶N}. IfA,B∈M, then(A:B) =∨{x ∈ L| x B¶A}. An L-module M is called a multiplication L-module if for every elementN∈M there exists an elementa∈Lsuch thatN =aIM see[2]. In this paper a lattice module M will be a multiplication lattice module, which is compactly generated with the largest elementI_Mcompact. A proper element N of M is said to be prime ifaX ¶Nimplies X ¶ _{N or aIM} ¶_{N that is a}¶ (N :IM)for everya ∈ L, X ∈M. If N is a prime element of M then(N :I_M)is prime element of L[4]. An elementN < I_M in M is said to be primary if aX ¶NimpliesX ¶N oranI_M¶N that isan¶(N:I_M)for some integer n. An element N of M is called a radical element if(N :IM) =

p

(N:IM). IfaN =0M impliesa =0 orN =0M for anya∈LandN∈M then M is called a torsion free L-module.

2. Residuation properties

We state some elementary properties of residuation in the following theorem.

Theorem 1. Let L be a multiplicative lattice and M be a multiplication lattice module over L.For x,y∈L and Z,A,B∈M , where(0_M:I_M)is a radical element. We have the following identities,

(i) x¶ _{y implies}(0_M :y)¶(0_M :x)and0_M:(0_M:x)¶_0M:(0_M:y)

(ii) x¶_0M:(0M :x)

(iv) (0_M :x) = (0_m:xn)for every n∈Z+

(v) 0_M:(0_M:x)_∧0_M:(0_M: y) =0_M:(0_M:x y) =0_M:[0_M :(x∧y)]

(vi) (0_M :a) =0_M implies(0_M:an) =0for every n∈Z+

(vii) x∨y =1implies(0_M:x)∨(0_M: y) =0_M :(x∧y) =0_M:x y (viii) For Z in M, Z¶0_M :(0_M:Z)

(ix) A¶_{B implies}(0M:B)¶(0M:A) (x) 0_M:[0_M:(0_M:A)] =0_M :A

(xi) 0_M:x I_M=0_M :xnI_M for some positive integer n.

We define, 0_{F M} =∨{X ∈M_∗|sX =0_Mfor somes ∈F}, where M_∗is the set of compact elements of M.

The proofs of the following theorems are simple

Theorem 2. Let F ⊆L be a filter of F(L∗)and let X be a compact element of M. Then X ¶0_{F M} if and only if sX =0_M for some s_∈F .

Theorem 3. For F∈F(L_∗),0_{F M} =_∨{(0_M:x)_|x∈F}. Theorem 4. For F1,F2∈F(L∗)

(i) F1⊆F2 implies0F1M¶0F2M.

(ii) 0_F

1M∧0F2M =0(F1

T

F2)M

3. Baer Elements

A study of Baer elements,∗-elements and closed elements carried out by D D Anderson,et al. [1]. We generalize these concepts for lattice modules.

Definition 1. An element A ∈ M is said to be Baer element if for x ∈ L∗, x IM ¶ A implies 0_M :(0_M:x I_M)¶_A.

Definition 2. An element A of M is said to be∗-element if A=0_{F M} for some filter F∈F(L_∗)such that zero does not belong to F.

Definition 3. An element A of M is said to be closed element if A=0_M:(0_M:A). The next result establishes the relation between closed element and Baer element. Theorem 5. Every closed element is a Baer element.

Proof. Let A be a closed element of M and x be a compact element ofL_∗such thatx I_M ¶_A.

Definition 4. An element P of M is called a minimal prime element over A∈M if A¶P and there is no other prime element Q of M such that A¶Q<P.

The following result gives the characterization of a minimal prime element over an ele-ment.

Theorem 6. Let a be proper element of L and P be a prime element of M with aIM ¶P. Then the following statements are equivalent,

(i) P is minimal prime element over aIM.

(ii) For each compact element x in L, x I_M¶P, there is compact element y in L such that y I_M P and xny IM ¶aIM =A for some positive integer n.

Proof. (i)⇒(ii)

Let P be a minimal prime overaIM and suppose x IM¶P. Let

S=_{xny _| y₆¶(P:I_M)andnis a positive integer_}.

It is clear that, S is a multiplicatively closed set. Suppose xny 6¶ aI_M for any integer n and for any y IMP, where y is compact in L. By the separation lemma (see[5]), there is a prime element (Q : I_M) of L such that (P : I_M) ¶ (Q : I_M) and t 6¶ (Q : I_M) for all t ∈S. Then we have(Q : I_M) ¶ (P : I_M) since otherwise xn(Q : I_M) ∈ S and xn(Q : I_M) 6¶ (Q : I_M) a contradiction. Hence(P: I_M) = (Q:I_M). It follows that P=Q (see[3]. But then for t _∈S, t¶_x ¶(P:IM) = (Q:IM)a contraduction.

(ii)⇒(i)

Suppose for any x in L, x I_M ¶ P, there is y in L such that y I_{M 6}¶ P and xny I_M ¶ aI_M for some positive integer n. Also suppose that there is a prime element Q of M withaIM¶Q<P. Choose, x I_M ¶P and x I_M 6¶Q. By hyphothesis, there is a compact element y in L such that y I_M 6¶ P and integer n such that xny I_M ¶ aI_M ¶Q. As x I_M Q, x (Q: I_M). Since Q is a prime element of M,(Q : IM) is also prime element of L (see [4]). Hence xn (Q : IM). Thus, xn 6¶ (Q : I_M) and y ¶6 (Q : I_M) where (Q : I_M)is a prime element of L, which is a contradiction.

In the next result, we prove the important property of a minimal prime element.

Theorem 7. Let M be an lattice module. Every minimal prime element of M is a∗-element where 0_{F M} is prime element.

compact element of L such that x I_M ¶0_{F M}. Then by Theorem 2,r x I_M =0_M for somer ∈F. So we have r x I_M ¶P and r I_M P. As P is prime, x I_M ¶ P and 0_{F M} ¶ Pwhich shows that P=0F M. Thus every minimal prime element of M is∗-element.

The relation between_∗-element and Baer element is proved in the next result. Theorem 8. Each∗-element of M is a Baer element.

Proof. Suppose an element A of M is∗-element. HenceA=0_{F M} for some filter F∈F(L_∗) such that 0∈/F. Letx ∈L_∗such thatx I_M¶A. Then we haver x I_M =0_Mthat isx I_M¶(0_M:r) for somer_∈F by Theorem 2. Therefore by(i)and(iii)of Theorem 1 we get

0_M:(0_M:x I_M)¶_0M:[0_M:(0_M:r)] = (0_M:r). Hence by Theorem 3, 0_M :(0_M : x I_M)¶ _∨

s∈F(0M :s) =0F M =A. This shows that A is a Baer element.

The next result we prove the existence of closed and Baer elements.

Theorem 9. Let M be multiplication lattice module. For any x ∈ L,(0M : x) is both Baer and closed element.

Proof. For an elementx ∈L_∗, let x I_M¶(0_M :x), then

0M:(0M:x IM)¶0M:[0M:(0M :x)] = (0M:x)

by(i)and(iii)of Theorem 1. Thus(0_M:x)is a Baer element. Again from(iii)of Theorem 1, (0_M :x) =0_M:(0_M:(0_M :x)). This shows that(0_M:x)is a closed element.

In the following theorem we prove the characterization of closed element in terms of Baer element.

Theorem 10. For a∈L_∗,aIM is closed if and only if aIM is a Baer element.

Proof. Let L_∗ be the set of all compact element of L andaI_M be a Baer element of M. We show thataI_M = 0_M :(0_M : aI_M). AsaI_M ¶ aI_M, we have[0_M :(0_M : aI_M)]¶ aI_M. But aIM(0M:aIM)¶0M impliesaIM ¶0M:(0M:aIM). Therefore 0M :(0M:aIM) =aIM. Thus aI_M is closed. The converse is proved in Theorem 5.

Theorem 11. For a nonzero compact element a in L,0_M :a=0_[a).

Proof. We note that F= [a) =_{z∈L_∗|z≥anfor somen∈Z+} ∈F(L∗)and

0_{F M} =∨{X ∈M_∗ | sX =0_M for somes∈ F}. Now let z be compact element of L such that z∈F∩ {0}. Thenz∈F andz=0. Asz∈F,z≥anfor somen∈Z₊. Hencea¶pz=0 which shows thata=0. This contradiction implies that 0_∈/ F. Now we show that 0_M:a=0_{F M}. As a is a compact element in L,a∈F. So we have 0_M :a¶_{0F M =∨{(0M:x)| x ∈F}. Let Z be}

a compact element in M andZ¶0_{F M}. Then by Theorem 2sZ=0_Mfor somes∈F. Sos≥an for somen∈Z+. We note that 0M:an=0M :a. Consequently, we haveanZ¶sZ=0M. This implies thatZ¶(0_M :an) = (0_M:a). Consequently, 0_F¶(0_M:a)and(0_M:a) =0_F.

Theorem 12. Suppose L has no divisors of zero then the element0_M is always a Baer, closed and

∗-element whereas1_M is Baer and closed.

Proof. Let x be a nonzero element of L. From Theorem 9,for any x∈L, 0_M :x is both Baer and closed and by Theorem 11 for a nonzero compact element x of L, 0_M:x =0_[x). To show that 0M a is Baer element,take x∈L∗such thatx IM¶0M. We have

0_M:(0_M:x IM)¶OM:(0M : 0M) =0M.

Hence 0_M is a Baer element. As 0_M =0_M :(0_M : 0_M), 0_M is closed. Every Baer element is a

∗-element. To show that 1M is a Baer element. Take anyx∈L∗such thatx IM¶1M. We have 0_M :(0_M : x I_M) = 0_M :[∨{a∈ L | a x I_M = 0_M}] =0_M : 0=1_M. So 1_M is a Baer element. Now 0_M :(0_M: 1_M) =0_M:[∨{a∈L|aI_M=0_M}] =1_M and 1_M is closed.

Remark 1. For defining the∗-element, the condition0∈/ F is necessary.

Suppose if possible X is a∗-element. Hence X =0_{F M}, for some filter F such that0∈/ F . Then we have X =_∨{(0M:r)|r ∈F}. Now0M : 0=∨{A∈M |0A=0M}=1M. Thus only1M will be a∗-element. Hence, for defining a∗-element we take F such that0∈/F .

Theorem 13. If{Aα}α is a family of Baer elements then∧_αAαis a Baer element.

Proof. Let x ∈ L_∗ such that x I_M ¶ _∧

αAα. Then for each α,x IM ¶ Aα. As eachAα is a Baer element, 0_M : (0_M : x I_M) ¶ A_α. Hence 0_M : (0_M : x I_M) ¶ _∧

αAα. Thus ∧αAα is a Baer element.

The next result we prove the relation between minimal prime element and Baer element. Theorem 14. If A is a meet of minimal prime elements then A is a Baer element.

Proof. From Theorem 7, every minimal prime element of M is a∗-element and by Theorem 8, each∗-element of M is a Baer element. From these two results,every minimal prime element is a Baer element. So meet of all minimal prime elements is a Baer element, by Theorem 13.

Theorem 15. If{Aα}αis a family of closed elements then∧_αAα is a closed element. Proof. We have∧

αAα¶Aα for eachα. As eachAα is a closed element we have

0_M :[0_M:(_∧Aα)]¶0M:(0M:Aα) =Aα. This gives 0M:[0M:(∧_αAα)]¶∧_αAα. Now let Z be an element of M such thatZ¶_∧

αAα. Then we haveZ¶0M:(0M:Z)¶0M :(0M :∧αAα), by(ix) of Theorem 1. This gives∧

αAα¶0M :[0M:(∧αAα)]. Thus we get 0M:[0M :(∧αAα)] =∧αAα. Here is an important property of largest element of M which is compact.

Proof. Suppose that 1_M is a∗-element. Then there exist some filter F ∈F(L_∗)such that 1_M =0_{F M}, where 0_∈/F. Then as 1_M is compact and 1_M =0_{F M} =_∨{(0_M:x)_|x _∈F_},

1M = (0M :x1)∨(0M :x2)∨. . .∨(0M :xn)for some x1,x2, . . . ,xn∈F. Consequently, as 1M is closed,

1_M=0_M:(0_M: 1_M) =0_M:[0_M :((0_M:x1)∨(0M :x2)∨. . .∨(0M:xn))] =0_M:[0_M:(0_M :x₁)∧0_M:(0_M:x₂)∧. . .∧0_M:(0_M:x_n)].

Therefore 1M =0M:[0M :(0M:(x1x2. . .xn)] =0M :(x1x2. . .xn), by(iii)and(v)of Theorem 1. This implies that x₁x₂. . .x_n=0. Sincex₁,x₂, . . . ,x_n are in F. We have 0=x₁x₂. . .x_n∈F. Which is a contradiction as 0∈/F.

The next result we prove the characterization of a Baer element. Theorem 17. The following statements are equivalent,

(i) An element A_∈M is a Baer element.

(ii) For any element x,y ∈L such that x is compact0_M:x I_M=0_M :y I_Mand x I_M ¶A implies y I_M ¶A.

(iii) For any element x,y∈L_∗, 0_M:x =0_M: y and x I_M ¶A implies y I_M¶A. Proof. (i)_⇒(ii)

Assume that A is a Baer element of M. Let x,y ∈L be such that x is compact, x I_M ¶_{A, and}

0_M :x I_M =0_M : y I_M. Then by Theorem 1, y I_M ¶0_M :(0_M : y I_M) =0_M :(0_M : x I_M)¶A, since A is a Baer element.

(ii)_⇒(iii) Obvious.

(iii)_⇒(i)

Assume that for any element x,y ∈ L_∗, 0M : x IM =0M : y IM and x IM ¶Aimplies y IM ¶A. We show thatA∈M is a Baer element. Let x∈L_∗be such that x I_M ¶A. We have

0_M :x I_M=0_M:[0_M:(0_M:x I_M)]. Hence by(iii), we have 0_M :(0_M :x I_M)¶A. Hence, A is a Baer element.

In the following theorem we prove the relation between Baer element of a lattice module and radical element of a multiplicative lattice.

Theorem 18. If A is Baer element of M then A:IM is a radical element.

Proof. Let A be Baer element of a lattice module M. We show that(A: I_M) =p(A:I_M). Assume that x is compact element such that xnI_M ¶Afor some positive integer n. We have 0M : x IM = 0M : xnIM, by (xii)of Theorem 1 and hence by above theorem x IM ¶ Athat is x ¶(A: I_M). Hencep(A:I_M)¶(A:I_M) and we havep(A:I_M) = (A: I_M) i.e.(A: I_M)is a radical element.

Proof. Let A be a Baer element and P be a minimal prime in M over A. Assume that 0_M :x =0_M:zfor some x,z∈Lsuch that x is compact and x I_M¶P. There exists a compact element y∈Lsuch that y IMPandxny IM¶A¶Pfor some positive integer n, by Theorem 14. Note that 0_M: y x = (0_M:x): y = (0_M:xn): y=0_M:xny=0_M :y xn=0_M : yz. As A is a Baer element. By Theorem 17, x y I_M ¶Aimplies yz I_M ¶A¶ P. Hencez I_M ¶ Pas P is prime. So again by Theorem 17, P is a Baer element.

The characterization of minimal prime element of M is proved in the next theorem. Theorem 20. Let L be a lattice module and P be a prime element of M. Then P is a minimal prime element if and only if for x∈L_∗, P contains precisely one of x I_M and0_M:x.

Proof. If part:

Assume that forx ∈L_∗,P contains precisely one ofx I_Mand 0_M:x. First assume that P contains x I_M. But 0_M :x P. Therefore there exists a compact element y in L such that y I_M¶0_M :x but y IM P. Thus x y IM ¶0M. This shows that for each compact element x in L,x IM ¶ P, there exist a compact element y in L such that y I_M P and x y I_M ¶ 0_M. By Theorem 6, it follows that P is a minimal prime element of M. Next assume that 0_M : x ¶ P but x I_M P. Let z be a compact element of L such thatz IM ¶(0M :x)¶ P. But x IMPand xz IM ¶0M. Consequently, by Theorem 6 P is a minimal prime element. Thus the condition is sufficient. Only if part:

Assume that P is a minimal prime element of M. Let x be a compact element of L. Suppose if possiblex I_M ¶P. Then by Theorem 6, there exist a compact element y in L such that y I_M P and xny I_M = 0_M for some positive integer n. Consequently, y I_M ¶0_M : xn = 0_M : x. This implies that 0M : x P. Now suppose if possible x IM P and 0M: x P. Then there exist a compact element y in L such that y I_M ¶0_M:x but y I_MP. Hence we have x y I_M ¶0_M and sox y I_M¶P. But x I_MPand y I_M P which contradicts the fact that P is prime element of M. This shows that P contains precisely one ofx IM and(0M:x).

The relation between∗-element of M and a minimal prime element over it is established in the next theorem.

Theorem 21. If A is a∗-element of M then every minimal prime over A is a minimal prime. Proof. Let P be a minimal prime element of M over A. We know by Theorem 8 and Theorem 18, a∗-element A is a Baer element and(A:I_M)is a radical element. Let x∈L_∗be such that x IM ¶P. But P is a minimal prime over A. Then by Theorem 2 there exists y ∈L∗ such that

y IM Pand xny IM ¶Ai.e. xny ¶A:IM. So xnyn ¶A:IM i.e. x y ¶

p

Remark 2. By Theorem 7, we infer that every minimal prime element is a∗-element and it is a Baer element. Therefore by Theorem 21, if A is the meet of all minimal prime elements containing it, A is a Baer element.

Notation: For a family{Aα}of Baer elements of L we define,

⊻_Aα=_∨{x IM,x∈L∗|0M :(x1∨x2. . .∨xn)IM¶0M :x IM, for some compact elements xjIM¶Aαj and some j=1, 2, . . . ,n}.

The important property of a family of Baer elements is established in the next theorem. Theorem 22. If {Aα}is a family of Baer elements of L,⊻Aα is the smallest Baer element greater than each A_α.

Proof. We first show that⊻A_α is a Baer element greater than eachA_α. Let x be a compact element of L such thatx IM¶ ⊻Aα. Then there exist compact elementsx1,x2, . . . ,xnsuch that 0_M :(x₁∨x₂∨. . .∨x_n)I_M ¶ _{0M : x IM and xjIM} ¶ _{Aαj j = 1, 2, . . . ,n. Next we show that}

0_M :(0_M:x I_M)¶ ⊻A_α. Let z be compact element in L such thatz I_M¶0_M:(0_M :x I_M). Then 0_M : z IM ≥ 0M : [0M : (0M : x IM)]. That is 0M : x IM ¶ 0M : z IM (by Theorem 1, (x)and (xi)). Therefore 0_M :(x₁∨x₂∨. . .∨x_n)I_M ¶_{0M :z IM. This implies thatz IM} ¶ ⊻_{Aα. Thus}

0_M :(0_M : x I_M)¶ ⊻A_α. This shows that⊻A_α is a Baer element. Let z be a compact element in L such thatz IM ¶Aα for someα. But 0M:z IM ¶0M :z IM. Thusz IM ¶ ⊻Aα. Hence each Aα¶ ⊻Aα. Let B be a Baer element such thatAα¶Bfor eachαand let x be a compact element in L such that 0_M:(x₁∨x₂∨. . .∨x_n)I_M¶0_M:x I_M for some compact elements x_jI_M ¶A_αj, j = 1, 2, . . . ,nso that x IM ¶ ⊻Aα. Note that B is a Baer element and the compact element

(x1∨x2∨. . .∨xn)IM ¶ B. Hence 0M : [0M : (x1∨x2∨. . .∨xn)IM] ¶ B. Again note that 0_M :(0_M : x I_M)¶0_M :[0_M :(x₁∨x₂∨. . .∨x_n)I_M] andx I_M ¶0_M :(0_M : x I_M). Therefore x IM ¶ B and hence ⊻Aα ¶ B. Consequently⊻Aα is the smallest Baer element greater than eachAα.

Theorem 23. For any proper element A∈M ,⊻_{0_M :(0_M :x IM)| x ∈L∗and x IM ¶A}is the smallest Baer element greater than A.

Proof. First we show that 0_M :(0_M : x I_M) is a Baer element i.e. we show that for any x ∈ L_∗, x I_M ¶ 0_M : (0_M : x I_M) implies 0_M : (0_M : x I_M) ¶ 0_M : (0_M : x I_M) which holds obviously. Hence by Theorem 22,B=⊻_{0M:(0M :x IM)| x∈L∗andx IM¶A}is the smallest Baer element containing each 0_M:(0_M :x I_M)forx I_M¶_{A. Let a compact element x in L be}

such thatx I_M¶A. Then we have x I_M¶0_M :(0_M:x I_M)¶B. ThusA¶B. Letz I_M be a Baer element in M such thatA¶_{z IM} and let y be compact element in L such that y IM ¶B. Then 0_M :(z₁∨z₂∨. . .∨z_n)I_M ¶_{0M : y IM, for some compact elements ziIM} ¶_{0M :}(0_M : x_iI_M), wherei=1, 2, . . . ,n. Thus 0_M :x_iI_M ¶0_M:z_iI_M for each i. This gives

0_M:(x₁∨x₂∨. . .∨x_n)I_M=0_M:x₁I_M∧0_M:x₂I_M∧. . . 0_M:x_nI_M

¶_{0M:z1IM∧0M:z2IM∧. . .∧0M:znIM}

Thus ifx =x₁∨x₂∨. . .∨x_nis compact element such thatx I_M= (x₁∨x₂∨. . .∨x_n)I_M¶A¶z I_M, we get 0_M:x I_M¶0_M: y I_M. Asz I_M is a Baer element we have

y I_M¶0_M :(0_M: y I_M)¶0_M:(0_M:x I_M)¶z I_M.

ThereforeB¶_{z IM}. This shows that⊻_{0_M :(0_M :x IM)|x ∈L∗andx IM¶A}is the smallest Baer element greater than A.

Notation : For a family{A_α}of closed elements of M we define, A▽B=∨{z I_M,z∈L_∗|0_M :(x∨y)I_M¶0_M:z I_M

for somex I_M¶Aand y I_M¶B}. Then we have the following important result. The property of closed elements is proved in the next theorem.

Theorem 24. If A and B are closed elements of M A▽B is the smallest closed element greater than A as well as B.

Proof. We show that A▽B is closed greater than A as well as B. Let C = A▽B. We always have C ¶ 0_M : (0_M : C) where C ∈ M. Let x be compact element in L such that x I_M¶0_M :(0_M:C). Then 0_M:C ¶0_M:x I_M. This implies that

0_M:(y∨z)IM¶0M:C¶0M:x IM

where y,z∈L_∗, y IM¶Aandz IM¶B. But y IM¶A▽B,z IM¶A▽B. Hence

0_M :(r∨s)I_M¶_{0M: y IMand 0M:}(u∨v)I_M¶_{0M:z IMwherer IM,uIM} ¶_{AandsIM,v IM}¶_B.

Therefore 0_M:(r∨s)I_M∧0_M:(u∨v)I_M ¶0_M: y I_M∧0_M :z I_M. Consequently 0_M:(r∨s∨u∨v)I_M¶0_M :(y∨z)I_M¶0_M:x I_M,

where(r∨u)I_M ¶Aand(s∨v)I_M ¶B. This implies thatx I_M¶C. Hence 0_M :(0_M:C)¶C. This gives 0_M:(0_M:C) =C and C is closed. As 0_M;sIM ¶0M :sIM for any element s in L, it follows thatA,B¶_{A▽B. Suppose that W is closed element such thatA,B}¶_{W and let x ∈L∗}

be such that 0_M:(u∨v)I_M¶0_M:x I_Mfor someuI_M ¶Aandv I_M¶B. Note that W is a closed element and(u_∨v)I_M ¶ W. Hence we have 0_M :[0_M :(u_∨v)I_M] ¶ 0_M :(0_M :W) =W. Again note that 0M :(0M : x IM)¶0M :[0M :(u∨v)IM]¶ W and x IM ¶0M :(0M : x IM). Therefore x I_M ¶W and henceA▽B¶W. Consequently, it proves thatA▽B is the smallest closed element greater than A as well as B.

Theorem 25. If A and B are closed elements of M then A▽B=0_M:[0_M:(A∨B)].

Proof. By Theorem 24, we haveA_∨B¶A_▽B. Hence 0_M:[0_M :(A_∨B)]¶A_▽BasA_▽Bis a closed element. Letx IM¶A▽B,x ∈L∗. Then 0M:(u∨v)IM¶0M:x IM, for someuIM¶A andv I_M ¶B. Consequently, we have

ACKNOWLEDGEMENTS The authors thank the readers of European Journal of Pure and Applied Mathematics, for making our journal successful. We dedicate this research article to

Prof Dr U Tekir & Prof Dr C Jayram.

References

[1] D.D. Anderson, C. Jayaram, and P.A. Phiri. Baer lattices. Acta Scientiarum Mathemati-carum, Vol 59. pps 61-74. 1994.

[2] F. Callialp and U. Tekir.Multiplication lattice modules, Iranian journal of science and tech-nology, Vol A4. pps 309-313. 2011.

[3] J. A. Johnson.a-adic completions of Noetherian lattice modules. Fundamenta Mathematica, Vol 66. pps. 341-371. 1970.

[4] A. KKhouja-Al Eaman.Maximal elements and prime elements in Lattice Modules. Damascus University Journal for Basic Sciences, Vol 19(2). 2003.