/compile/output.dvi EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 8, No. 1, 2015, 15-25 ISSN 1307-5543 – www.ejpam.com Characterization of Prime Ideals in (Z+,≤D) Sankar Sagi College of Applied Sciences, Sohar, Ministry of Higher Education, Sultanate of Oman Abstract. A convolution is a mapping C of the set Z+ of positive integers into the set P (Z+) of all subsets of Z+ such that, for any n ∈ Z+ , each member of C (n) is a divisor of n. If D(n) is the set of all divisors of n, for any n, then D is called the Dirichlet’s convolution. Corresponding to any general convolution C , we can define a binary relation ≤C on Z+ by “m ≤C n if and only if m ∈ C (n)”. It is well known that Z+ has the structure of a distributive lattice with respect to the division order. The division ordering is precisely the partial ordering ≤D induced by the Dirichlet’s convolution D. In this paper, we present a characterization for the prime ideals in (Z+,≤D) , where D is the Dirichlet’s convolution. 2010 Mathematics Subject Classifications: 06B10,11A99 Key Words and Phrases: Poset, Lattice, semi lattice, Convolution, ideal 1. Introduction A Convolution is a mapping C : Z+ −→P (Z+) such that C (n) is a set of positive divisors on n, n ∈ C (n) andC (n) = ⋃ m∈C (n) C (m), for any n ∈ Z+. Popular examples are the Dirichlet’s convolution D and the Unitary convolution U defined respectively by D(n) = The set of all positive divisors of n and U (n) = The set of Unitary divisors of n for any n ∈ Z+. If C is a convolution, then the binary relation ≤C on Z+, defined by, m≤C n if and only if m ∈ C (n), is a partial order on Z+ and is called the partial order induced by C [2]. It is well known that the Dirichlet’s convolution induces the division order onZ+ with respect to whichZ+ becomes a distributive lattice, where, for any a, b ∈ Z+, the greatest common divisor(GCD) and the Email address: sagi_sankar@yahoo.co.in http://www.ejpam.com 15 c© 2015 EJPAM All rights reserved. S. Sagi / Eur. J. Pure Appl. Math, 8 (2015), 15-25 16 least common multiple(LCM) of a and b are respectively the greatest lower bound(glb) and the least upper bound(lub) of a and b . In fact, with respect to the division order, the lattice Z+ satisfies the infinite join distributive law given by � a ∨ ( ∧ i∈I bi) = ∧ i∈I (a ∨ bi) � for any a ∈ Z+ and {bi}i∈I ⊆ Z +. In this paper, we discuss various aspects of ideals and filters in (Z+,≤C) and eventually present a characterization of prime ideals in (Z+,≤D) where D is the Dirichlet’s convolution Actually a general convolution may not induce a lattice structure on Z+. However , most of the convolutions we are considering induce a meet semi lattice structure on Z+. For this reason, we first consider a general semi lattice and study it’s ideals and later extend these to (Z+,≤D). 2. Preliminaries Let us recall that a partial order on a non-empty set X is defined as a binary relation ≤ on X which is reflexive (a ≤ a), transitive (a ≤ b, b ≤ c =⇒ a ≤ c) and antisymmetric (a ≤ b, b ≤ a =⇒ a = b) and that a pair (X ,≤) is called a partially ordered set(poset) if X is a non-empty set and ≤ is a partial order on X . For any A ⊆ X and x ∈ X , x is called a lower(upper) bound of A if x ≤ a(respectively a ≤ x) for all a ∈ A. We have the usual notations of the greatest lower bound(glb) and least upper bound(lub) of A in X . If A is a finite subset {a1, a2, · · · , an}, the glb of A(lub of A) is denoted by a1 ∧ a2 ∧ · · · ∧ an or n ∧ i=1 ai (respectively by a1∨ a2∨· · ·∨ an or n ∨ i=1 ai). A partially ordered set (X ,≤) is called a meet semi lattice if a∧ b (=glb{a, b}) exists for all a and b ∈ X . (X ,≤) is called a join semi lattice if a∨ b (=lub{a, b}) exists for all a and b ∈ X . A poset (X ,≤) is called a lattice if it is both a meet and join semi lattice. Equivalently, lattice can also be defined as an algebraic system (X ,∧,∨), where ∧ and ∨ are binary operations which are associative, commutative and idempotent and satisfying the absorption laws, namely a ∧ (a ∨ b) = a = a ∨ (a ∧ b) for all a, b ∈ X ; in this case the partial order ≤ on X is such that a ∧ b and a ∨ b are respectively the glb and lub of {a, b}. The algebraic operations ∧ and ∨ and the partial order ≤ are related by (a = a ∧ b⇐⇒ a ≤ b⇐⇒ a ∨ b = b) . Throughout the paper, Z+ and N denote the set of positive integers and the set of non- negative integers respectively. Definition 1. A mappingC :Z+ −→P (Z+) is called a convolution if the following are satisfied for any n ∈ Z+. (1). C (n) is a set of positive divisors of n (2). n ∈ C (n) S. Sagi / Eur. J. Pure Appl. Math, 8 (2015), 15-25 17 (3). C (n) = ⋃ m∈C (n) C (m). Definition 2. For any convolution C and m and n ∈ Z+, we define � m≤ n if and only if m ∈ C (n) � Then ≤C is a partial order on Z+ and is called the partial order induced by C on Z+. In fact, for any mapping C : Z+ −→ P (Z+) such that each member of C (n) is a divisor of n, ≤C is a partial order on Z+ if and only if C is a convolution, as defined above [1, 4]. Definition 3. Let C be a convolution and p a prime number. Define a relation ≤p C on the set N of non-negative integers by � a ≤p C b if and only if pa ∈ C (pb) � for any a and b ∈ N . It can be easily verified that ≤p C is a partial order on N , for each prime p. The following is a direct verification. Theorem 1. Let C be a convolution. (1). If (Z+,≤C ) is a meet(join) semilattice, then so is (N ,≤p C ) for each prime p. (2). If (Z+,≤C ) is a lattice, then so is (N ,≤p C ) for each prime p. 3. Ideals in (Z +,≤D) Recall that most of the convolutions like Dirichlet’s convolution, Unitary convolution and k-free convolution induce meet semi lattice structure on Z+ [3]. For this reason we study ideals in a general meet semi lattice and later study ideals in the lattice structure Z+ induced by the division ordering /. The division ordering / is precisely the partial ordering≤D induced by the Dirichlet’s convolution D. Throughout this section, unless otherwise stated, by a semi lattice we mean a meet semi lattice only. Definition 4. Let (X ,≤) be a poset. A non-empty subset I of X is called an initial segment if a ∈ I , x ∈ X and x ≤ a =⇒ x ∈ I . Definition 5. Let (S,∧) be a semi lattice. A non-empty subset I of S is called an ideal of S if the following are satisfied (1). x ∈ S and x ≤ a ∈ I =⇒ x ∈ I (2). For any a and b ∈ I , there exists c ∈ I such that a ≤ c and b ≤ c S. Sagi / Eur. J. Pure Appl. Math, 8 (2015), 15-25 18 Definition 6. Let (S,∧) be a semi lattice and a ∈ S. Then the set (a] := {x ∈ S|x ≤ a}= {y ∧ a|y ∈ S} is an ideal of S and is called the Principal ideal generated by a in S. Note that (a] is the smallest ideal of S containing a. Now, we present the following Theorem 2. Let a and b be elements of a meet semi lattice (S,∧). Then the following are equiv- alent to each other. (1). There exists smallest ideal of S containing a and b. (2). The intersection of all ideals of S containing a and b is again an ideal of S. (3). a and b have least upper bound in S. Proof. (1)⇐⇒ (2) : is trivial. (1) =⇒ (3) : Let I be the smallest ideal of S containing a and b. Then, there exists x ∈ I such that a ≤ x and b ≤ x Therefore x is an upper bound of a and b. If y is any other upper bound of a and b, then (y] is an ideal of S containing a and b and hence I ⊆ (y]. Since x ∈ I , we get that x ∈ (y] and therefore x ≤ y . Thus x is the least upper bound of a and b. (3) =⇒ (1) : Let a∨ b be the least upper bound of a and b. Then a ≤ a∨ b and b ≤ a∨ b and hence (a ∨ b] is an ideal containing a and b. If I is any ideal containing a and b, then there exists x ∈ I such that a ≤ x and b ≤ x and hence a ∨ b ≤ x so that a∨ b ∈ I and (a∨ b] ⊆ I . Thus (a∨ b] is the smallest ideal of S containing a and b. Although the intersection of an arbitrary class of ideals need not be an ideal, a finite inter- section is always an ideal. Theorem 3. Let (S,∧) be a semi lattice and I (S) the set of all ideals of S. Then (I (S),∩) is a semilattice and a 7→ (a] is an embedding of (S,∧) onto (I (S),∩). Proof. By the above theorem, it follows that (I (S),∩) is a semi lattice.Also, for any a and b in S, we have (a]∩ (b] = (a ∧ b] and (a] ⊆ (b]⇐⇒ a ∈ (b]⇐⇒ a ≤ b Therefore a 7→ (a] is an embedding of S into I (S). Theorem 4. A semi lattice (S,∧) is a lattice if and only if I (S) is a lattice and, in this case, a 7→ (a] is an embedding of the lattice S into the lattice I (S). S. Sagi / Eur. J. Pure Appl. Math, 8 (2015), 15-25 19 Proof. It is well known that the set I (S) of ideals of a lattice (S,∧,∨) is again a lattice in which, I ∧ J = I ∩ J and I ∨ J = {x ∈ S|x ≤ a ∧ b, for some a ∈ I and b ∈ J} for any ideals I and J , in this case, (a]∨ (b] = (a ∨ b] for any a and b in S, so that a 7→ (a] is an embedding of lattices. Conversely, suppose that I (S) is a lattice. Let a and b ∈ S and I be the least upper bound of (a] and (b] in I (S). Then I is the smallest ideal containing a and b and hence by Theorem 2, a ∨ b exists in S. Therefore S is a lattice. For a lattice (L,∧,∨), any ideal of the semi lattice (L,∧) turns out to be the usual ideal of the lattice (L,∧,∨). Definition 7. Let (S,∧) be a semi lattice. A non-empty subset F of S is called filter of S if, for any a, b ∈ S, a ∧ b ∈ F⇔ a ∈ F and b ∈ F Theorem 5. Let (S,∧) be a semi lattice and P a proper ideal of S. Then the following are equivalent to each other (1). For any elements a and b in S, a ∧ b ∈ P =⇒ a ∈ P or b ∈ P (2). For any ideals I and J of S, I ∩ J ⊆ P =⇒ I ⊆ P or J ⊆ P (3). S − P is a filter of S. Proof. (1) =⇒ (2): Let I and J be ideals of S. Suppose that I 6⊆ P and J 6⊆ P. Then there exist a ∈ I and b ∈ J such that a /∈ P and b /∈ P. Then,by (1), a∧ b /∈ P. But a∧ b ≤ a ∈ I and a ∧ b ≤ b ∈ J and hence a ∧ b ∈ I ∩ J . Therefore I ∩ J 6⊆ P. (2) =⇒ (3): If a ≤ b and a ∈ S − P, then clearly b ∈ S − P. Also, a and b ∈ S − P =⇒a /∈ P and b /∈ P =⇒(a] 6⊆ P and (b] 6⊆ P =⇒(a ∧ b] = (a]∩ (b] 6⊆ P =⇒x /∈ P for some x ≤ a ∧ b =⇒x ≤ a ∧ b and x ∈ S − P =⇒a ∧ b ∈ S − P Thus S − P is a filter of S. (3) =⇒ (1): For any a and b ∈ S, S. Sagi / Eur. J. Pure Appl. Math, 8 (2015), 15-25 20 a /∈ P and b /∈ P =⇒a and b ∈ S − P =⇒a ∧ b ∈ S − P =⇒a ∧ b /∈ P Definition 8. Any proper ideal P of a semi lattice (S,∧) is said to be a prime ideal if any one (and hence all) of the conditions in Theorem 5 is satisfied. 4. Prime Ideals in (Z +,≤D) Now we shall turn our attention to the particular case of the lattice structure onZ+ induced by the division ordering / and study the ideals and prime ideals of Z+. The division ordering is precisely the partial ordering ≤D induced by the Dirichlet’s convolution D. First we observe that � θ : (Z+,/) −→ ( ∑ P N ,≤) � is an order isomorphism where θ is defined by � θ (a)(p) = The largest n ∈ N such that pn divides a , for any a ∈ Z+ and p ∈ P � and � ∑ P N � = { f :P −→ N | f (p) = 0 for all but finite p}. Here P stands for the set of primes and N stands for the set of non-negative integers. Definition 9. Adjoin an external element∞ to N and extend the usual ordering ≤ on N to N ∪ {∞} by defining a <∞ for all a ∈ N . We shall denote N ∪ {∞} together with this extended usual order by N ∞ . Theorem 6. Let α :P −→N ∞ be a mapping and define Iα = {n ∈ Z +|θ (n)(p)≤ α(p) for all p ∈ P } Then Iα is an ideal of (Z+,/) and every ideal of (Z+,/) is of the form Iα for some mapping α :P −→N ∞ Proof. Since no prime divides the integer 1, we get that θ (1)(p) = 0 ≤ α(p) for all p ∈ P and hence 1 ∈ Iα. Therefore Iα is a non-empty subset of Z+ . m and n ∈ Iα =⇒θ (m)(p)≤ α(p) and θ (n)(p)≤ α(p) for all p ∈ P =⇒θ (m∨ n)(p) = Max{θ (m)(p),θ (n)(p)} ≤ α(p) for all p ∈ P S. Sagi / Eur. J. Pure Appl. Math, 8 (2015), 15-25 21 =⇒m∨ n ∈ Iα and m≤D n ∈ Iα =⇒θ (m)(p)≤ θ (n)(p)≤ α(p) for all p ∈ P =⇒θ (m)(p)≤ α(p) for all p ∈ P =⇒m ∈ Iα. Thus Iα is an ideal of (Z+,/). Conversely suppose that I is any ideal of (Z+,/). Define α :P −→N ∞ by α(p) = Sup{θ (n)(p)|n ∈ I} for any p ∈ P Note that α(p) is either a non-negative integer or∞, for any p ∈ P . Therefore α is a mapping of P into N ∞. n ∈ I =⇒θ (n)(p)≤ α(p) for all p ∈ P =⇒n ∈ Iα Therefore I ⊆ Iα. On the other hand, suppose n ∈ Iα. Then θ (n)(p) ≤ α(p) for all p ∈ P . Since θ (n) ∈ ∑ P N , |θ (n)| is finite. If |θ (n)|= φ, then n= 1 ∈ I . Suppose |θ (n)| is non-empty. Let |θ (n)| = {p1, p2 · · · , pr}. Then θ (n)(p) = 0 for all p 6= pi , 1 ≤ i ≤ r and θ (n)(pi) ∈ N . Now, for each 1 ≤ i ≤ r, θ (n)(pi) ≤ α(pi) = Sup{θ (m)(pi)|m ∈ I} and hence there exists mi ∈ I such that θ (n)(pi)≤ θ (m)(pi). Now, put m= m1 ∨m2 ∨ · · · ∨mr , then m ∈ I and θ (n)(pi)≤ Max{θ (m1)(pi), . . . ,θ (mi)(pi)}= θ (m)(pi) for all 1 ≤ i ≤ r. Also, since θ (n)(p) = 0 for all p 6= pi , we get that θ (n)(p) ≤ θ (m)(p) for all p ∈ P so that n≤D m ∈ I and therefore n ∈ I . Therefore Iα ⊆ I . Thus I = Iα. Note that, if α is the constant map 0 defined by α(p) = 0 for all p ∈ P , then Iα = {1} and that, if α is the constant map∞, then Iα = Z +. Definition 10. For any mappings α and β from P into N ∞, define α≤ β if and only if α(p)≤ β(p) for all p ∈ P . Thus ≤ is a partial order on (N ∞)P . Theorem 7. The map α 7→ Iα is an order isomorphism of the poset ((N ∞)P ,≤), onto the poset (I (Z+),⊆) of all ideals of (Z+,/). Proof. Let α and β : P 7→ N ∞ be any mappings. Clearly, α ≤ β ⇒ Iα ⊆ Iβ . On the other hand, suppose that Iα ⊆ Iβ . We shall prove that α(p) ≤ β(p) for all p ∈ P so that α ≤ β . To prove this, let us fix p ∈ P . If β(p) =∞ or α(p) = 0, trivially α(p) ≤ β(p). Therefore, we can assume that β(p)<∞ and α(p)> 0. Consider n= pβ(p)+1. Then θ (n)(p) = β(p) + 1 6≤ β(p). S. Sagi / Eur. J. Pure Appl. Math, 8 (2015), 15-25 22 and hence n /∈ Iβ . Since Iα ⊆ Iβ , n /∈ Iα and therefore θ (n)(q) 6≤ α(q) for some q ∈ P . But θ (n)(q) = 0 for all q 6= p. Thus β(p) + 1=θ (n)(p) 6≤ α(p) α(p)<β(p) + 1. Therefore α(p) ≤ β(p). This is true for all p ∈ P . Thus α ≤ β . Also α 7→ Iα is a surjection. Thus α 7→ Iα is an order isomorphism of ((N ∞)P ,≤), onto (I (Z+),⊆). Corollary 1. For any α and β :P →N ∞, Iα ∩ Iβ = Iα∧β . and Iα ∪ Iβ = Iα∨β . where α∧ β and α∨ β are point-wise g.l.b and l.u.b of α and β . First we state the following two theorems from “Lattice Structures on Z+ induced by con- volutions” [3]. Theorem 8. Let C be a convolution which is closed under finite intersections and ≤C be the partial order on Z+ induced by C . Then (Z+,≤C ) is a lattice if and only if it is directed above. Theorem 9. Let C be a convolution. (1). If (Z+,≤C ) is a meet(join) semilattice, then so is (N ,≤p C ) for each prime p (2). If (Z+,≤C ) is a lattice, then so is (N ,≤p C ) for each prime p. Theorem 10. Let C be a multiplicative convolution such that (Z+,/) is a meet semi lattice. For any α :P →N ∞, let Iα = {n ∈ Z +|θ (n)(p)≤PC α(p) for all p ∈ P }. Then the following are equivalent to each other. (1). Iα is an ideal of (Z+,≤C ) for any α :P →N ∞. (2). (Z+,≤C ) is directed below (3). (Z+,≤C ) is a lattice. Proof. (2)⇔ (3) follows from Theorem 8 (1)⇒ (2) : Let α :P →N ∞ be defined by α(p) =∞ for all p ∈ P . Then Iα = {n ∈ Z +|θ (n)(p)≤PC α(p) =∞ for all p ∈ P } and hence, by (1), Z+ is an ideal of (Z+,≤C ) which implies that (Z+,≤C ) is directed above. (3)⇒ (1) : From (3) and Theorems 8 and 9, it follows that (N ,≤p C ) is a lattice for each p ∈ P S. Sagi / Eur. J. Pure Appl. Math, 8 (2015), 15-25 23 and θ (m ∨ n)(p) = θ (m)(p) ∨ θ (n)(p) in (N ,≤p C ) for any m and n ∈ Z+ and p ∈ P . Let α :P →N ∞ be any mapping. Then, for any m and n ∈ Z+, m≤C n ∈ Iα =⇒θ (m)(p)≤ p C θ (n)(p)≤ p C α(p) for all p ∈ P . =⇒θ (m)(p)≤p C for all p ∈ P . =⇒m ∈ Iα. and m and n ∈ Iα =⇒θ (m)(p)≤ p C α(p) and θ (n)(p)≤p C α(p) for all p ∈ P . =⇒θ (m)(p)∨ θ (n)(p)≤p C α(p) for all p ∈ P . =⇒m∨ n ∈ Iα. Therefore Iα is an ideal of (Z+,≤C ). Now, we have the following Theorems which characterize the prime ideals of the lattice (Z+,≤D) where D is the Dirichlet’s convolution. Theorem 11. Let α : P → N ∞ be a mapping and Iα is an ideal of (Z+,≤D) defined by Iα = {n ∈ Z +|θ (n)(p)≤ α(p) for all p ∈ P }. Then the following are equivalent to each other. (1). Iα is a prime ideal of (Z+,≤D). (2). α(p) 6=∞ for some p ∈ P and for any β and γ :P −→N ∞, β ∧ γ≤ α=⇒ β ≤ α or γ≤ α. (3). There exists unique p ∈ P such that α(p) 6=∞ and α(q) =∞ for all q 6= p ∈ P . Proof. (1) =⇒ (2) follows from Theorem 7, in which we have proved that β 7→ Iβ is an isomorphism of the lattice ((N ∞)P ,≤) onto the lattice of ideals of (Z+,≤D) from the fact that Iβ ∩ Iγ = Iβ∧γ for any β and γ : P −→ N ∞. If α(p) =∞ for all p ∈ P , then, since θ (n)(p) ∈ N for all n ∈ Z+ and p ∈ P , Iα = {n ∈ Z +|θ (n)(p)<∞}= Z+ which is a contradiction to the fact that every prime ideal is a proper ideal. Thus α(p) 6=∞ for some p ∈ P . (2) =⇒ (3): Suppose that α satisfies (2). Fix p ∈ P such that α(p) 6=∞. Then α(p) ∈ N . Now, define β and γ :P −→N ∞ by β(q) = ¨ 0 if q = p ∞ if q 6= p S. Sagi / Eur. J. Pure Appl. Math, 8 (2015), 15-25 24 and γ(q) = ¨ ∞ if q = p 0 if q 6= p for any q ∈ P . Then, (β ∧ γ)(q) = β(q)∧ γ(q) = 0≤ α(q) for all q ∈ P and hence β ∧γ≤ α. Since α(p) 6=∞ and γ(p) =∞, (γ)(p) 6≤ α(p) and hence γ 6≤ α. Therefore, by (2), β ≤ α and hence ∞ = β(q)≤ α(q) for all q 6= p. Therefore q(p) =∞ for all q 6= p in P . This also implies the uniqueness of p. (3) =⇒ (1): Let p ∈ P such that α(p) 6=∞ and α(q) =∞ for all q 6= p ∈ P . Then Iα is a proper ideal of (Z+,≤D). Let J and K be any ideals of (Z+,≤D) such that J ∩ K ⊆ Iα. Then there exists β and γ : P −→ N ∞ such that J = Iβ and K = Iγ. Now, Iβ∧γ = Iβ ∩ Iγ = J ∩ K ⊆ Iα and hence β ∧ γ≤ α so that Min{β(p),γ(p)} = (β ∧ γ)(p)≤ α(p). Therefore β(p)≤ α(p) or γ(p)≤ α(p). Since α(q) =∞ for all q 6= p, it follows that β ≤ α or γ ≤ α and hence Iβ ⊆ Iα or Iγ ⊆ Iα. Therefore J ⊆ Iα or K ⊆ Iα. Thus Iα is a prime ideal of (Z+,≤D). Definition 11. For any prime number p and a ∈ N , define Ip,a = {n ∈ Z +|θ (n)(p)≤ a}. Then Ip,a is an ideal of (Z+,≤D). In fact Ip,a = Iα, where α :P −→N ∞ is defined by α(q) = ¨ a if q = p ∞ if q 6= p Note that Ip,a = {n ∈ Z +|pa+1 does not divide n}. Theorem 12. An ideal of (Z+,≤D) is prime if and only if it is of the form Ip,a for some p ∈ P and a ∈ N . Proof. Let I be an ideal of (Z+,≤D). Then I = Iα for some mapping α :P −→N ∞. Now, by Theorem 11, I is prime⇐⇒ there exists p ∈ P such that α(p) 6=∞ and α(q) =∞ for all q 6= p and I = Iα⇐⇒ I = Ip,a, where a = α(p). Theorem 13. For any p and q ∈ P and a and b ∈ N , Ip,a ⊆ Iq,b⇐⇒ p = q and a ≤ b REFERENCES 25 Proof. If p = q and a ≤ b, then n ∈ Ip,q =⇒θ (n)(p)≤ a ≤ b =⇒θ (n)(q)≤ b =⇒n ∈ Iq,b and hence Ip,a ⊆ Iq,b. Conversely suppose that Ip,a ⊆ Iq,b. If p 6= q, then θ (qb+1)(p) = 0≤ a and hence qb+1 ∈ Ip,a ⊆ Iq,b so that θ (qb+1)(b)≤ b, which is a contradiction. Therefore p = q. Now, since θ (pa)(p) = a, pa ∈ Ip,a ⊆ Iq,b and hence a = θ (pa)(q) ≤ b. Thus p = q and a ≤ b. The following are immediate consequences of Theorems 11,12 and 13. Corollary 2. For each p ∈ P , let Pp = {Ip,a|a ∈ N }. Then the following hold. (1). Pp is a chain of prime ideals of (Z+,≤D) for each p ∈ P . (2). Pp ∩Pq = φ for all p 6= q ∈ P . (3). ⋃ p∈P Pp is the set of all prime ideals of (Z+,≤D). Corollary 3. I is a minimal prime ideal of (Z+,≤D) if and only if I = Ip,0 = {n ∈ Z +|p does not divide n} for some p ∈ P . References [1] S. Sagi. Lattice Theory of Convolutions, Ph.D. Thesis, Andhra University, Waltair, Visakha- patnam, India, 2010. [2] U.M. Swamy, G.C. Rao, V. S. Ramaiah. On a conjecture in a ring of arithmetic functions. Indian Journal of Pure and Applied Mathematics, 14(12), 1519-1530. 1983. [3] U.M. Swamy, S. Sagi. Lattice Structures onZ+ induced by convolutions. European Jounal of Pure and Applied Mathematics,4(4), 424-434. 2011. [4] U.M. Swamy, S. Sagi. Partial Orders induced by Convolutions. International Journal of Mathematics and Soft Computing, 2(1), 2011, 25-33.