/compile/output.dvi EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 8, No. 3, 2015, 332-342 ISSN 1307-5543 – www.ejpam.com Baer Elements In Lattice Modules C S Manjarekar 1, U N Kandale 2,∗ 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 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 then p a = ∨{x ∈ L | xn ¶ a, n ∈ Z+}. An element a ∈ L is called a radical element if a = p a. An element a ∈ L is called compact if a ¶ ∨ α bα 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 shall 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, (i) x , y ∈ F implies x y ∈ F (ii) x ∈ F, x ¶ y implies y ∈ F . ∗Corresponding author. Email addresses: csmanjrekar@yahoo.co.in (C Manjarekar), ujwalabiraje@gmail.com (U Kandale) http://www.ejpam.com 332 c© 2015 EJPAM All rights reserved. C Manjarekar, U Kandale / Eur. J. Pure Appl. Math, 8 (2015), 332-342 333 Let F(L∗) denote the set of all filters of L. For a nonempty subset {Fα} ⊆ F(L∗), define ⋒Fα = {x ∈ L∗ | x ≥ f1 f2 · · · 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 theroretic ⋂ 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, for s ∈ 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 aB where a ∈ L and B ∈ M which satisfies the following properties, (i) (∨ α aα)A= ∨ α aαA ∀aα ∈ L, A∈ M (ii) a(∨ α Aα) = ∨ α aAα ∀a ∈ L, Aα ∈ M (iii) (ab)A= a(bA) ∀a, b ∈ L, A∈ M (iv) 1B = B (v) 0B = 0M for all a, aα, b ∈ L and A,Aα ∈ M , where 1 is the supremum of L and 0 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 . . .. Let M be a 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 see [2]. In this paper a lattice module M will be a multiplication lattice module, which is compactly generated with the largest element IM compact. 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 [4]. An element N < IM in M is said to be primary if aX ¶ N implies X ¶ N or an IM ¶ N that is an ¶ (N : IM ) for some integer n. An element N of M is called a radical element if (N : IM ) = p (N : IM ). If aN = 0M implies a = 0 or N = 0M for any a ∈ L and N ∈ 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 (0M : IM ) is a radical element. We have 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) C Manjarekar, U Kandale / Eur. J. Pure Appl. Math, 8 (2015), 332-342 334 (iv) (0M : x) = (0m : xn) for every n ∈ Z+ (v) 0M : (0M : x)∧ 0M : (0M : y) = 0M : (0M : x y) = 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 : x y (viii) For Z in M, Z ¶ 0M : (0M : Z) (ix) A¶ B implies (0M : B)¶ (0M : A) (x) 0M : [0M : (0M : A)] = 0M : A (xi) 0M : x IM = 0M : xn IM for some positive integer n. We define, 0F M = ∨{X ∈ M∗ | sX = 0M for some s ∈ 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 ¶ 0F M if and only if sX = 0M for some s ∈ F. Theorem 3. For F ∈ F(L∗), 0F M = ∨{(0M : x) | x ∈ F}. Theorem 4. For F1, F2 ∈ F(L∗) (i) F1 ⊆ F2 implies 0F1M ¶ 0F2M . (ii) 0F1M ∧ 0F2M = 0(F1 ⋂ 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 0M : (0M : x IM )¶ A. Definition 2. An element A of M is said to be ∗-element if A= 0F 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= 0M : (0M : 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 of L∗ such that x IM ¶ A. Then 0M : (0M : x IM ) ¶ 0M : (0M : A) = A as A is a closed. This shows that A is a Baer element. C Manjarekar, U Kandale / Eur. J. Pure Appl. Math, 8 (2015), 332-342 335 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 IM ¶ P, there is compact element y in L such that y IM � P and xn y IM ¶ aIM = A for some positive integer n. Proof. (i)⇒ (ii) Let P be a minimal prime over aIM and suppose x IM ¶ P. Let S = {xn y | y 6¶ (P : IM ) and n is a positive integer }. It is clear that, S is a multiplicatively closed set. Suppose xn y 6¶ aIM for any integer n and for any y IM � P, where y is compact in L. By the separation lemma (see [5]), there is a prime element (Q : IM ) of L such that (P : IM ) ¶ (Q : IM ) and t 6¶ (Q : IM ) for all t ∈ S. Then we have (Q : IM ) ¶ (P : IM ) since otherwise xn(Q : IM ) ∈ S and xn(Q : IM ) 6¶ (Q : IM ) a contradiction. Hence (P : IM ) = (Q : IM ). 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 IM ¶ P, there is y in L such that y IM 6¶ P and xn y IM ¶ aIM for some positive integer n. Also suppose that there is a prime element Q of M with aIM ¶Q < P. Choose, x IM ¶ P and x IM 6¶ Q. By hyphothesis, there is a compact element y in L such that y IM 6¶ P and integer n such that xn y IM ¶ aIM ¶ Q. As x IM � Q, x � (Q : IM ). 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 : IM ) and y 6¶ (Q : IM ) where (Q : IM ) 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 0F M is prime element. Proof. Let p be a minimal prime element of M. Define the set F = {x ∈ L∗ | x IM � P}. We first show that F is a filter of F(L∗). Let x and y be compact element of L such that x , y ∈ F . So x IM � P and y IM � P. As P is prime, x y IM � P. This shows that x y ∈ F . Now let x ∈ F and x ¶ y . Hence x IM � P implies y IM � P and y ∈ F . If 0 ∈ F then we have 0IM � P that is 0M � P a contradction. Thus F ∈ F(L∗) and 0 /∈ F . Now we show that P = 0F M . Let x be a compact element of L such that x IM ¶ P. By Theorem 6 it follows that there exist a compact element y ∈ L such that y IM � P and xn y IM = 0M for some positive integer n. We have y ∈ F and xn IM ¶ 0F M . As 0F M is prime element, so x IM ¶ 0F M implies P ¶ 0F M . Now let x be a C Manjarekar, U Kandale / Eur. J. Pure Appl. Math, 8 (2015), 332-342 336 compact element of L such that x IM ¶ 0F M . Then by Theorem 2, r x IM = 0M for some r ∈ F . So we have r x IM ¶ P and r IM � P. As P is prime, x IM ¶ P and 0F M ¶ P which 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. Hence A= 0F M for some filter F ∈ F(L∗) such that 0 /∈ F . Let x ∈ L∗ such that x IM ¶ A. Then we have r x IM = 0M that is x IM ¶ (0M : r) for some r ∈ F by Theorem 2. Therefore by (i) and (iii) of Theorem 1 we get 0M : (0M : x IM )¶ 0M : [0M : (0M : r)] = (0M : r). Hence by Theorem 3, 0M : (0M : x IM ) ¶ ∨ 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 element x ∈ L∗, let x IM ¶ (0M : x), then 0M : (0M : x IM )¶ 0M : [0M : (0M : x)] = (0M : x) by (i) and (iii) of Theorem 1. Thus (0M : x) is a Baer element. Again from (iii) of Theorem 1, (0M : x) = 0M : (0M : (0M : x)). This shows that (0M : 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 and aIM be a Baer element of M. We show that aIM = 0M : (0M : aIM ). As aIM ¶ aIM , we have [0M : (0M : aIM )] ¶ aIM . But aIM (0M : aIM ) ¶ 0M implies aIM ¶ 0M : (0M : aIM ). Therefore 0M : (0M : aIM ) = aIM . Thus aIM is closed. The converse is proved in Theorem 5. Theorem 11. For a nonzero compact element a in L, 0M : a = 0[a). Proof. We note that F = [a) = {z ∈ L∗ | z ≥ an for some n ∈ Z+} ∈ F(L∗) and 0F M = ∨{X ∈ M∗ | sX = 0M for some s ∈ F}. Now let z be compact element of L such that z ∈ F ∩{0}. Then z ∈ F and z = 0. As z ∈ F, z ≥ an for some n ∈ Z+. Hence a ¶ p z = 0 which shows that a = 0. This contradiction implies that 0 /∈ F . Now we show that 0M : a = 0F M . As a is a compact element in L, a ∈ F . So we have 0M : a ¶ 0F M = ∨{(0M : x) | x ∈ F}. Let Z be a compact element in M and Z ¶ 0F M . Then by Theorem 2 sZ = 0M for some s ∈ F . So s ≥ an for some n ∈ Z+. We note that 0M : an = 0M : a. Consequently, we have anZ ¶ sZ = 0M . This implies that Z ¶ (0M : an) = (0M : a). Consequently, 0F ¶ (0M : a) and (0M : a) = 0F . The following theorem establishes the property of Baer, closed and ∗-element. C Manjarekar, U Kandale / Eur. J. Pure Appl. Math, 8 (2015), 332-342 337 Theorem 12. Suppose L has no divisors of zero then the element 0M is always a Baer, closed and ∗-element whereas 1M is Baer and closed. Proof. Let x be a nonzero element of L. From Theorem 9,for any x ∈ L, 0M : x is both Baer and closed and by Theorem 11 for a nonzero compact element x of L, 0M : x = 0[x). To show that 0M a is Baer element,take x ∈ L∗ such that x IM ¶ 0M . We have 0M : (0M : x IM )¶ OM : (0M : 0M ) = 0M . Hence 0M is a Baer element. As 0M = 0M : (0M : 0M ), 0M is closed. Every Baer element is a ∗-element. To show that 1M is a Baer element. Take any x ∈ L∗ such that x IM ¶ 1M . We have 0M : (0M : x IM ) = 0M : [∨{a ∈ L | ax IM = 0M}] = 0M : 0 = 1M . So 1M is a Baer element. Now 0M : (0M : 1M ) = 0M : [∨{a ∈ L | aIM = 0M}] = 1M and 1M is closed. Remark 1. For defining the ∗-element, the condition 0 /∈ F is necessary. Suppose if possible X is a ∗-element. Hence X = 0F M , for some filter F such that 0 /∈ F. Then we have X = ∨{(0M : r) | r ∈ F}. Now 0M : 0 = ∨{A ∈ M | 0A= 0M} = 1M . Thus only 1M will be a ∗-element. Hence, for defining a ∗-element we take F such that 0 /∈ F. Theorem 13. If {Aα}α is a family of Baer elements then ∧ α Aα is a Baer element. Proof. Let x ∈ L∗ such that x IM ¶ ∧ α Aα. Then for each α, x IM ¶ Aα. As each Aα is a Baer element, 0M : (0M : x IM ) ¶ Aα. Hence 0M : (0M : x IM ) ¶ ∧ α Aα. Thus ∧ α Aα 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}α is a family of closed elements then ∧ α Aα is a closed element. Proof. We have ∧ α Aα ¶ Aα for each α. As each Aα is a closed element we have 0M : [0M : (∧Aα)]¶ 0M : (0M : Aα) = Aα. This gives 0M : [0M : (∧ α Aα)]¶ ∧ α Aα. Now let Z be an element of M such that Z ¶ ∧ α Aα. Then we have Z ¶ 0M : (0M : Z) ¶ 0M : (0M : ∧ α Aα), by (ix) of Theorem 1. This gives ∧ α Aα ¶ 0M : [0M : (∧ α Aα)]. Thus we get 0M : [0M : (∧ α Aα)] = ∧ α Aα. Here is an important property of largest element of M which is compact. Theorem 16. 1M is never a ∗-element where 1M is compact and M is torsion free L-module. C Manjarekar, U Kandale / Eur. J. Pure Appl. Math, 8 (2015), 332-342 338 Proof. Suppose that 1M is a ∗-element. Then there exist some filter F ∈ F(L∗) such that 1M = 0F M , where 0 /∈ F . Then as 1M is compact and 1M = 0F M = ∨{(0M : x) | x ∈ F}, 1M = (0M : x1)∨ (0M : x2)∨ . . .∨ (0M : xn) for some x1, x2, . . . , xn ∈ F . Consequently, as 1M is closed, 1M =0M : (0M : 1M ) = 0M : [0M : ((0M : x1)∨ (0M : x2)∨ . . .∨ (0M : xn))] =0M : [0M : (0M : x1)∧ 0M : (0M : x2)∧ . . .∧ 0M : (0M : xn)]. Therefore 1M = 0M : [0M : (0M : (x1 x2 . . . xn)] = 0M : (x1 x2 . . . xn), by (iii) and (v) of Theorem 1. This implies that x1 x2 . . . xn = 0. Since x1, x2, . . . , xn are in F. We have 0 = x1 x2 . . . xn ∈ 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 compact 0M : x IM = 0M : y IM and x IM ¶ A implies y IM ¶ A. (iii) For any element x , y ∈ L∗, 0M : x = 0M : y and x IM ¶ A implies y IM ¶ A. Proof. (i)⇒ (ii) Assume that A is a Baer element of M. Let x , y ∈ L be such that x is compact, x IM ¶ A, and 0M : x IM = 0M : y IM . Then by Theorem 1, y IM ¶ 0M : (0M : y IM ) = 0M : (0M : x IM ) ¶ 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 ¶ A implies y IM ¶ A. We show that A∈ M is a Baer element. Let x ∈ L∗ be such that x IM ¶ A. We have 0M : x IM = 0M : [0M : (0M : x IM )]. Hence by (iii), we have 0M : (0M : x IM ) ¶ 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 : IM ) = p (A : IM ). Assume that x is compact element such that xn IM ¶ A for some positive integer n. We have 0M : x IM = 0M : xn IM , by (xii) of Theorem 1 and hence by above theorem x IM ¶ A that is x ¶ (A : IM ). Hence p (A : IM ) ¶ (A : IM ) and we have p (A : IM ) = (A : IM ) i.e.(A : IM ) is a radical element. Theorem 19. If A is a Baer element then every minimal prime element over A is a Baer element. C Manjarekar, U Kandale / Eur. J. Pure Appl. Math, 8 (2015), 332-342 339 Proof. Let A be a Baer element and P be a minimal prime in M over A. Assume that 0M : x = 0M : z for some x , z ∈ L such that x is compact and x IM ¶ P. There exists a compact element y ∈ L such that y IM � P and xn y IM ¶ A¶ P for some positive integer n, by Theorem 14. Note that 0M : y x = (0M : x) : y = (0M : xn) : y = 0M : xn y = 0M : y xn = 0M : yz. As A is a Baer element. By Theorem 17, x y IM ¶ A implies yzIM ¶ A¶ P. Hence zIM ¶ P as 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 IM and 0M : x. Proof. If part: Assume that for x ∈ L∗,P contains precisely one of x IM and 0M : x . First assume that P contains x IM . But 0M : x � P. Therefore there exists a compact element y in L such that y IM ¶ 0M : 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 IM � P and x y IM ¶ 0M . By Theorem 6, it follows that P is a minimal prime element of M. Next assume that 0M : x ¶ P but x IM � P. Let z be a compact element of L such that zIM ¶ (0M : x) ¶ P. But x IM � P and xzIM ¶ 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 possible x IM ¶ P. Then by Theorem 6, there exist a compact element y in L such that y IM � P and xn y IM = 0M for some positive integer n. Consequently, y IM ¶ 0M : xn = 0M : 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 IM ¶ 0M : x but y IM � P. Hence we have x y IM ¶ 0M and so x y IM ¶ P. But x IM � P and y IM � P which contradicts the fact that P is prime element of M. This shows that P contains precisely one of x 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 : IM ) 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 � P and xn y IM ¶ A i.e. xn y ¶ A : IM . So xn yn ¶ A : IM i.e. x y ¶ p (A : IM ) = (A : IM ). By hyphothesis, x y is compact and x y IM ¶ A= 0F M , for some filter F of L∗ such that 0 /∈ F . Hence x y IM d = 0M for some d ∈ F . We show that there is no compact element x in F such that x IM ¶ P. Suppose there is compact element z in L such that zIM ¶ P and z ∈ F . Then by Theorem 3, 0M : z ¶ 0F = A¶ P. This contradict the fact that P contains precisely one of zIM and 0M : z where z ∈ L∗. Hence there is no compact element x in F such that x IM ¶ P. This implies that dIM 6¶ P. As P is prime, dIM 6¶ P and y IM 6¶ P implies yd IM 6¶ P. Thus x yd IM = 0M ¶ P and yd IM 6¶ P. Therefore by Theorem 6, P is minimal prime. C Manjarekar, U Kandale / Eur. J. Pure Appl. Math, 8 (2015), 332-342 340 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 x j IM ¶ Aα j 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 Aα. Proof. We first show that ⊻Aα is a Baer element greater than each Aα. Let x be a compact element of L such that x IM ¶ ⊻Aα. Then there exist compact elements x1, x2, . . . , xn such that 0M : (x1 ∨ x2 ∨ . . . ∨ xn)IM ¶ 0M : x IM and x j IM ¶ Aα j j = 1,2, . . . , n. Next we show that 0M : (0M : x IM )¶ ⊻Aα. Let z be compact element in L such that zIM ¶ 0M : (0M : x IM ). Then 0M : zIM ≥ 0M : [0M : (0M : x IM )]. That is 0M : x IM ¶ 0M : zIM (by Theorem 1, (x) and (xi)). Therefore 0M : (x1 ∨ x2 ∨ . . . ∨ xn)IM ¶ 0M : zIM . This implies that zIM ¶ ⊻Aα. Thus 0M : (0M : x IM ) ¶ ⊻Aα. This shows that ⊻Aα is a Baer element. Let z be a compact element in L such that zIM ¶ Aα for some α. But 0M : zIM ¶ 0M : zIM . Thus zIM ¶ ⊻Aα. Hence each Aα ¶ ⊻Aα. Let B be a Baer element such that Aα ¶ B for each α and let x be a compact element in L such that 0M : (x1 ∨ x2 ∨ . . .∨ xn)IM ¶ 0M : x IM for some compact elements x j IM ¶ Aα j , j = 1,2, . . . , n so 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 0M : (0M : x IM ) ¶ 0M : [0M : (x1 ∨ x2 ∨ . . . ∨ xn)IM ] and x IM ¶ 0M : (0M : x IM ). Therefore x IM ¶ B and hence ⊻Aα ¶ B. Consequently ⊻Aα is the smallest Baer element greater than each Aα. Theorem 23. For any proper element A ∈ M, ⊻{0M : (0M : x IM ) | x ∈ L∗ and x IM ¶ A} is the smallest Baer element greater than A. Proof. First we show that 0M : (0M : x IM ) is a Baer element i.e. we show that for any x ∈ L∗, x IM ¶ 0M : (0M : x IM ) implies 0M : (0M : x IM ) ¶ 0M : (0M : x IM ) which holds obviously. Hence by Theorem 22, B = ⊻{0M : (0M : x IM ) | x ∈ L∗ and x IM ¶ A} is the smallest Baer element containing each 0M : (0M : x IM ) for x IM ¶ A. Let a compact element x in L be such that x IM ¶ A. Then we have x IM ¶ 0M : (0M : x IM ) ¶ B. Thus A¶ B. Let zIM be a Baer element in M such that A¶ zIM and let y be compact element in L such that y IM ¶ B. Then 0M : (z1 ∨ z2 ∨ . . . ∨ zn)IM ¶ 0M : y IM , for some compact elements zi IM ¶ 0M : (0M : x i IM ), where i = 1,2, . . . , n. Thus 0M : x i IM ¶ 0M : zi IM for each i. This gives 0M :(x1 ∨ x2 ∨ . . .∨ xn)IM = 0M : x1 IM ∧ 0M : x2 IM ∧ . . . 0M : xn IM ¶0M : z1 IM ∧ 0M : z2 IM ∧ . . .∧ 0M : zn IM =0M : (z1 ∨ z2 ∨ . . .∨ zn)IM ¶ 0M : y IM . C Manjarekar, U Kandale / Eur. J. Pure Appl. Math, 8 (2015), 332-342 341 Thus if x = x1∨x2∨. . .∨xn is compact element such that x IM = (x1∨x2∨. . .∨xn)IM ¶ A¶ zIM , we get 0M : x IM ¶ 0M : y IM . As zIM is a Baer element we have y IM ¶ 0M : (0M : y IM )¶ 0M : (0M : x IM )¶ zIM . Therefore B ¶ zIM . This shows that ⊻{0M : (0M : x IM ) | x ∈ L∗ and x IM ¶ A} is the smallest Baer element greater than A. Notation : For a family {Aα} of closed elements of M we define, A▽ B = ∨{zIM , z ∈ L∗ | 0M : (x ∨ y)IM ¶ 0M : zIM for some x IM ¶ A and y IM ¶ 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 ¶ 0M : (0M : C) where C ∈ M . Let x be compact element in L such that x IM ¶ 0M : (0M : C). Then 0M : C ¶ 0M : x IM . This implies that 0M : (y ∨ z)IM ¶ 0M : C ¶ 0M : x IM where y, z ∈ L∗, y IM ¶ A and zIM ¶ B. But y IM ¶ A▽ B, zIM ¶ A▽ B. Hence 0M : (r∨s)IM ¶ 0M : y IM and 0M : (u∨v)IM ¶ 0M : zIM where r IM ,uIM ¶ A and sIM , vIM ¶ B. Therefore 0M : (r ∨ s)IM ∧ 0M : (u∨ v)IM ¶ 0M : y IM ∧ 0M : zIM . Consequently 0M : (r ∨ s ∨ u∨ v)IM ¶ 0M : (y ∨ z)IM ¶ 0M : x IM , where (r ∨ u)IM ¶ A and (s ∨ v)IM ¶ B. This implies that x IM ¶ C . Hence 0M : (0M : C) ¶ C . This gives 0M : (0M : C) = C and C is closed. As 0M ; sIM ¶ 0M : sIM for any element s in L, it follows that A, B ¶ A▽ B. Suppose that W is closed element such that A, B ¶W and let x ∈ L∗ be such that 0M : (u∨ v)IM ¶ 0M : x IM for some uIM ¶ A and vIM ¶ B. Note that W is a closed element and (u ∨ v)IM ¶ W . Hence we have 0M : [0M : (u ∨ v)IM ] ¶ 0M : (0M : W ) = W . Again note that 0M : (0M : x IM ) ¶ 0M : [0M : (u ∨ v)IM ] ¶ W and x IM ¶ 0M : (0M : x IM ). Therefore x IM ¶W and hence A▽ B ¶W . Consequently, it proves that A▽ 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 = 0M : [0M : (A∨ B)]. Proof. By Theorem 24, we have A∨B ¶ A▽B. Hence 0M : [0M : (A∨B)]¶ A▽B as A▽B is a closed element. Let x IM ¶ A▽B, x ∈ L∗. Then 0M : (u∨ v)IM ¶ 0M : x IM , for some uIM ¶ A and vIM ¶ B. Consequently, we have x IM ¶ 0M : (0M : x IM )¶ 0M : [0M : (u∨ v)IM ]¶ 0M : [0M : (A∨ B)]. Hence A▽ B ¶ (0M : 0M : (A∨ B)). Thus A▽ B = 0M : [0M : (A∨ B)]. REFERENCES 342 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. 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