Strongly Prime Ideals and Primal Ideals in Posets

KYUNGPOOK Math. J. 56(2016), 727-735 http://dx.doi.org/10.5666/KMJ.2016.56.3.727 pISSN 1225-6951 eISSN 0454-8124 c Kyungpook Mathematical Journal ⃝ Strongly Prime Ideals and Primal Ideals in Posets Catherine Grace John and Balasubramanian Elavarasan∗ Department of Mathematics, Karunya University, Coimbatore-641114, Tamil Nadu, India. e-mail : [email protected] and [email protected] Abstract. In this paper, we study and establish some interesting results of ideals in a poset. It is shown that for a nonzero ideal I of a poset P , there are at most two strongly prime ideals of P that are minimal over I. Also, we study the notion of primal ideals in a poset and the relationship among the primal ideals and strongly prime ideals is considered. 1. Introduction Throughout this paper (P , ≤) denotes a poset with smallest element 0. For basic terminology and notation for posets, we refer [9] and [6]. For M ⊆ P, let L(M ) = {x ∈ P : x ≤ m for all m ∈ M } denote the lower cone of M in P and dually, let U (M ) = {x ∈ P : m ≤ x for all m ∈ M } be the upper cone of M in P. Let A, B ⊆ P , we shall write L(A, B) instead of L(A∪B) and dually for the upper cones. If M = {x1 , x2 , ..., xn } is finite, then we use the notation L(x1 , x2 , ..., xn ) instead of L({x1 , x2 , ..., xn })(and dually). It is clear that for any subset A of P , we have A ⊆ L(U (A)) and A ⊆ U (L(A)). If A ⊆ B, then L(B) ⊆ L(A) and U (B) ⊆ U (A). Moreover, LU L(A) = L(A) and U LU (A) = U (A). Following [10], a non-empty subset I of P is called a semi-ideal if b ∈ I and a ≤ b, then a ∈ I. A subset I of P is called an ideal if a, b ∈ I implies L(U ((a, b)) ⊆ I[9]. Following [8], for any subset X of P , [X] is the smallest ideal of P containing X. If X = {b}, then L(b) is called the principle ideal of P generated by b. A proper semi-ideal (ideal) I of P is called prime if L(a, b) ⊆ I implies that either a ∈ I or b ∈ I [6]. An ideal I of a poset P is called semi-prime if L(a, b) ⊆ I and L(a, c) ⊆ I together imply L(a, U (b, c))) ⊆ I[9]. Following [3], an ideal I of P is called strongly prime if L(A∗ , B ∗ ) ⊆ I implies that either A ⊆ I or B ⊆ I for different proper ideals A, B of P, where A∗ = A\{0}. A * Corresponding Author. Received July 23, 2015; revised March 21, 2016; accepted April 7, 2016. 2010 Mathematics Subject Classification: 06A06. Key words and phrases: Poset, ideals, strongly prime ideal, primal ideal, minimal prime ideal. 727 728 Catherine Grace John and B. Elavarasan non-empty subset M of P is called an m-system if for any x1 , x2 ∈ M , there exists t ∈ L(x1 , x2 ) such that t ∈ M . Following [1], a non-empty subset M of P is called a strongly m-system if A ∩ M ̸= ∅ and B ∩ M ̸= ∅ imply L(A∗ , B ∗ ) ∩ M ̸= ∅ for different proper ideals A, B of P . It is clear that an ideal I of P is strongly prime if and only if P/I is a strongly m- system of P . Also every strongly m-system of P is an m-system. Following [3], an ideal I of P is called strongly semi-prime if L(A∗ , B ∗ ) ⊆ I and L(A∗ , C ∗ ) ⊆ I together imply L(A∗ , U (B ∗ , C ∗ )) ⊆ I for any different ideals A, B and C of P. For any semi-ideal I of∩ P and a subset A of P , we < a, I >[3]. If A = {x}, define < A, I >= {z ∈ P : L(a, z) ⊆ I for all a ∈ A} = a∈A then we write < x, I > instead of < {x}, I >. For any ideal I of P , a strongly prime ideal Q of P is said to be a minimal strongly prime ideal of I if I ⊆ Q and there exists no strongly prime ideal R of P such that I ⊂ R ⊂ Q. The set of all strongly prime ideals of P is denoted by Sspec(P ) and the set of minimal strongly prime ideals of P is denoted by Smin(P ). For any ideal I of P , P (I) and SP (I) denotes the intersection of all prime semi-ideals and strongly prime ideals of P containing I. It is clear from Theorem 6 of [6] and Example 1.1 of [2] that P (I) = I and SP (I) ̸= I for any ideal I of P . Following [1], let I be a semi-ideal of P. Then I is said to have (*) condition if whenever L(A, B) ⊆ I, we have A ⊆< B, I > for any subsets A and B of P. 2. Main Results Theorem 2.1. Let M be a nonempty strongly m-system of P and J be an ideal of P with J ∩ M = ∅. Then J is contained in a strongly prime ideal I of P with I ∩ M = ∅. Proof. Let S = {K : K is an ideal of P with K ∩ M = ∅}. Then S ̸= ∅ and by Zorn’s lemma, there exists a maximal element I ∈ S with I ∩ M = ∅. Let A and B be ideals of P with L(A∗ , B ∗ ) ⊆ I and suppose that A * I and B * I. Then there exists x ∈ A\I and y ∈ B\I such that I ⊂ I ∪ {x} ⊆ [I ∪ {x}] and I ⊂ I ∪ {y} ⊆ [I ∪ {y}], which imply [I ∪ {x}] ∩ M ̸= ∅ and [I ∪ {y}] ∩ M ̸= ∅. Since M is strongly m-system, we have L([I ∪ {x}]∗ , [I ∪ {y}]∗ ) ∩ M ̸= ∅. But L([I ∪ {x}]∗ , [I ∪ {y}]∗ ) ⊆ L([I ∪ {x}]∗ ) ⊆ L(I ∗ ) ⊆ I, which implies I ∩ M ̸= ∅, a contradiction.  Theorem 2.2. Let I and J be ideals of P with {0} ̸= J ⊆ I. Then the following are equivalent. (i) I is a minimal strongly prime ideal of J. (ii) For each x ∈ I, there exists t ∈ U (x) and y ∈ P \I such that L(L(t)∗ , L(y)∗ ) ⊆ J. (iii) If I has (*) condition, then for any x ∈ I, we have < x, J >* I. Proof. (i) ⇒ (ii) Let I be a minimal strongly prime ideal of J. Suppose that there exists x ∈ I such that L(L(ti )∗ , L(yj )∗ ) * J for all ti ∈ U (x) and yj ∈ P \I. Let Strongly Prime Ideals and Primal Ideals in Posets 729 M = {aij : aij ∈ L(L(ti )∗ , L(yj )∗ )\J for ti ∈ U (x) and yj ∈ P \I}. Then M ̸= ∅. For any ideals A, B of P , let A ∩ M ̸= ∅ and B ∩ M ̸= ∅. Then there exists a ∈ A and b ∈ B such that a, b ∈ M . Let t ∈ L(A∗ , B ∗ ). Then t ∈ L(a, b). Since a, b ∈ M , we have a ∈ L(L(ti )∗ , L(yj )∗ )\J and b ∈ L(L(tk )∗ , L(yl )∗ )\J for some ti , tk ∈ U (x) and yj , yl ∈ P \I, which imply t ∈ L(L(ti )∗ , L(yj )∗ ) with t ∈ / J. Indeed, if t ∈ J, then a ∈ L(L(ti )∗ ) ⊆ L(t) ⊆ J, a contradiction. So M is a strongly m-system of P . Since M ∩ J = ∅ and by Theorem , there exists a strongly prime ideal I1 of P containing J with I1 ∩ M = ∅. If x ∈ I1 , then L(L(x)∗ , L(yi )∗ ) ⊆ I1 for every yi ∈ P \I. But there exists q ∈ L(L(ti )∗ , L(yj )∗ )\J with q ∈ M , which implies q ∈ L(L(ti )∗ , L(yj )∗ ) ⊆ L(L(x)∗ , L(yi )∗ ) ⊆ I1 and I1 ∩ M ̸= ∅, a contradiction. So x∈ / I1 . Let i1 ∈ I1 and suppose i1 ∈ / I. Then i1 ∈ P \I and L(L(x)∗ , L(i1 )∗ ) ⊆ I1 . ∗ ∗ But L(L(ti ) , L(i1 ) ) * J, which implies I1 ∩ M ̸= ∅, a contradiction. Thus I1 ⊂ I, which is again a contradiction to the minimality of I. (ii) ⇒ (i) Let I1 be a strongly prime ideal of P with J ⊆ I1 ⊆ I. Let x ∈ I. Then there exists y ∈ P \I and t ∈ U (x) such that L(L(t)∗ , L(y)∗ ) ⊆ J ⊆ I1 . Since y∈ / I1 , we have L(t) ⊆ I1 , which implies x ≤ t ∈ I1 . Thus I ⊆ I1 and hence I is a minimal strongly prime ideal of J. (i) ⇒ (iii) Let x ∈ I. Then by (ii), there exists y ∈ / I and t ∈ U (x) such that L(L(t)∗ , L(y)∗ ) ⊆ J. Since J satisfies (*) condition, we have y ∈ L(y)∗ ⊆< L(t)∗ , J >⊆< x, J >, which implies < x, J >* I. (iii) ⇒ (i) Let Q be a strongly prime ideal of P such that J ⊆ Q ⊂ I and x ∈ I\Q. Then < x, J >* I. So there exists y ∈< x, J > \I such that L(L(x)∗ , L(y)∗ ) ⊆ L(x, y) ⊆ J ⊆ Q. Since L(x) * Q, we have y ∈ Q, a contradiction.  The following example shows that the condition J ̸= {0} is not superficial in Theorem 2.2. Example 2.3. Consider P = {0, 1, 2, 3} and define a relation ≤ on P as follows. b 3 b 2 b 1 b 0 Then (P, ≤) is a poset and I = {0, 1} is a minimal strongly prime ideal of J = {0}. But for 1 ∈ I\J, there is no y ∈ P \I and t ∈ U (1) such that L(L(t)∗ , L(y)∗ ) ⊆ J.  Following [6], a semi-ideal I of P is called n-prime if for pairwise distinct elements x1 , x2 , x3 , ..., xn ∈ P , if L(x1 , x2 , x3 , ..., xn ) ⊆ I, then at least (n − 1) of n subsets L(x2 , x3 , x4 , ..., xn ), L(x1 , x3 , x4 , ..., xn ), ..., L(x1 , x2 , x3 , ..., xn−1 ) is a subset 730 Catherine Grace John and B. Elavarasan of I. One can define in an obvious way the concept of n-strongly prime ideal of P for n ≥ 2. As a special case, if I is strongly prime, then I is a 2-strongly prime ideal. It might be hard to extend the results for n > 2. It is proved by the following theorem that n-primeness can not be generalized for n > 2. Theorem 2.4. Let I be an ideal of P . Then I has the following property that for n > 2, if pairwise distinct ideals A1 , A2 , ..., An of P with L(A∗1 , A∗2 , ..., A∗n ) ⊆ I, then at least (n−1) of n subsets L(A∗2 , A∗3 , ..., A∗n ), L(A∗1 , A∗3 , ..., A∗n ), ..., L(A∗1 , A∗2 , ..., A∗n−1 ) are subsets of I. Proof. Let A1 , A2 , ..., An be distinct ideals of P with L(A∗1 , A∗2 , ..., A∗n ) ⊆ I. Suppose that L(A∗1 , A∗2 , ..., A∗n−1 ) * I. Then we now prove L(A∗2 , A∗3 , ..., A∗n ), L(A∗1 , A∗3 , ..., A∗n ), ..., L(A∗1 , A∗2 , ..., A∗n−2 , A∗n ) are subsets of I. Since L(A∗1 , A∗2 , ..., A∗n−1 ) * I, then there exists t ∈ L(A∗1 , A∗2 , ..., A∗n−1 )\I such that L(t) ⊆ L(A∗1 , A∗2 , ..., A∗n−1 ), which n−1 ∪ A∗i , A∗n ) ⊆ implies L(t, A∗n ) ⊆ I. So for each j ∈ {1, 2, 3, ..., n − 1}, we have L( L(t, A∗n ) ⊆ I. i=1,i̸=j  In the generalization of n-primeness in posets, by the above theorem, we get that the cases n = 2 and n ≥ 3 are substantially different. Hence a 2-strongly prime ideal only exists in P . If n > 2, then every ideal of P is n-strongly prime. Lemma 2.5. Let I be a semi-prime ideal of P with (*) condition. Then < C ∗ , I > is a strongly prime ideal of P for any ideal C of P. Let A, B and C be ideals of P with L(A∗ , B ∗ ) ⊆< C ∗ , I >. Then L(A , B , C ∗ ) ⊆ I. By Theorem 2.4, we have A ⊆< C ∗ , I > or B ⊆< C ∗ , I >. /,  Proof. ∗ ∗ Theorem 2.6. Let J ̸= {0} be an ideal of P . Then there are at most two strongly prime ideals of P that are minimal over J. Proof. Suppose that I1 , I2 and I3 are three pairwise distinct strongly prime ideals of P that are minimal over J. Then there exist x1 ∈ I1 \I2 and x2 ∈ I2 \I1 . Since x1 ∈ I1 and by Theorem 2.2, there exist c2 ∈ / I1 and t1 ∈ U (x1 ) such that L(L(t1 )∗ , L(c2 )∗ ) ⊆ J. Also for x2 ∈ I2 , there exists c1 ∈ / I2 and t2 ∈ U (x2 ) such that L(L(t2 )∗ , L(c1 )∗ ) ⊆ J. Then L(L(t2 )∗ , L(c1 )∗ ) ⊆< x1 , J >, which implies L(L(t2 )∗ , L(c1 )∗ , L(x1 )∗ ) ⊆ J. Suppose L(L(x1 )∗ , L(c1 )∗ ) ⊆ J. Then L(L(x1 )∗ , L(c1 )∗ ) ⊆ I2 . Since I2 is a strongly prime ideal of P , we have c1 ∈ L(c1 ) ⊆ I2 or x1 ∈ L(x1 ) ⊆ I2 , a contradiction. By Theorem 2.4, we have L(L(t2 )∗ , L(c1 )∗ ) ⊆ J and L(L(t2 )∗ , L(x1 )∗ ) ⊆ J. Clearly I1 * I2 ∪ I3 , I2 * I1 ∪ I3 and I3 * I1 ∪ I2 . Indeed, if I1 ⊆ I2 ∪ I3 , then I1∗ ⊆ I2∗ ∪ I3∗ , which implies L(I2∗ , I3∗ ) ⊆ I1 . Since I1 is strongly prime, we have I2 ⊆ I1 or I3 ⊆ I1 , a contradiction. So we can choose y1 ∈ I1 \(I2 ∪ I3 ); y2 ∈ I2 \(I1 ∪ I3 ); y3 ∈ I3 \(I2 ∪ I1 ). By the above argument, we have L(L(t2 )∗ , L(y1 )∗ ) ⊆ J ⊆ I3 for some t2 ∈ U (y2 ). Since I3 is strongly prime, we have y1 ∈ I3 or y2 ≤ t2 ∈ I3 , a contradiction. So there are at most two strongly prime ideal that are minimal over J.  Strongly Prime Ideals and Primal Ideals in Posets 731 Theorem 2.7. Let J ̸= {0} be an ideal of P . Then at least one of the following statement must hold. (i) SP (J) = I1 is a strongly prime ideal of P . (ii) SP (J) = I1 ∩ I2 , where I1 and I2 are the distinct strongly prime ideal of P that are minimal over J. If J satisfies (*) condition, then L(I1∗ , I2∗ ) ⊆ J. Proof. By Theorem 2.6, we have SP (J) = I1 is a strongly prime ideal of P or SP (J) = I1 ∩I2 , where I1 and I2 are the distinct strongly prime ideals of P that are minimal over J. Since I1 and I2 are distinct, there exists x1 ∈ I1 \I2 and x2 ∈ I2 \I1 . By Theorem 2.2, there exist c2 ∈ / I1 and t1 ∈ U (x) such that L(L(t1 )∗ , L(c2 )∗ ) ⊆ J and there exist c1 ∈ / I2 and t2 ∈ U (x) such that L(L(t2 )∗ , L(c1 )∗ ) ⊆ J. Then L(L(t2 )∗ , L(c1 )∗ ) ⊆< x1 , J >, which implies L(L(t2 )∗ , L(c1 )∗ , L(x1 )∗ ) ⊆ J. Suppose L(L(x1 )∗ , L(c1 )∗ ) ⊆ J. Then L(L(x1 )∗ , L(c1 )∗ ) ⊆ I2 , so c1 ∈ L(c1 ) ⊆ I2 or x1 ∈ L(x1 ) ⊆ I2 , a contradiction. By Theorem 2.4, we have L(L(t2 )∗ , L(c1 )∗ ) ⊆ J and L(L(t2 )∗ , L(x1 )∗ ) ⊆ J, which imply x1 ∈ L(x1 )∗ ⊆< L(t2 )∗ , J >⊆< x2 , J > . Thus L(x1 , x2 ) ⊆ J and hence L(I1∗ , I2∗ ) ⊆ J.  Theorem 2.8. Let J ̸= {0} be a semi-prime ideal of P such that J ̸= SP (J) = I is a strongly prime ideal of P and J satisfies (*) condition. Then for each x ∈ I\J and I ⊆< x, J >, we have < x, J > is a strongly prime of P . Furthermore < y, J >⊆< x, J > or < x, J >⊆< y, J > for every x, y ∈ I\J. Proof. Let x ∈ I\J and I ⊂< x, J > with L(A∗ , B ∗ ) ⊆< x, J > for different proper ideals A, B of P . If L(A∗ , B ∗ ) ⊆ I, then A ⊆< x, J > or B ⊆< x, J > . Let L(A∗ , B ∗ ) * I. Since L(A∗ , B ∗ ) ⊆< x, J >, we have L(A∗ , B ∗ , L(x)∗ ) ⊆ J. Then by Theorem 2.4, we have L(A∗ , L(x)∗ ) ⊆ J and L(B ∗ , L(x)∗ ) ⊆ J, which imply A ⊆< x, J > and B ⊆< x, J >. So < x, J > is a strongly prime ideal of P . Let x, y ∈ I\J. If < x, J ≯⊂< y, J >, then there exists z ∈< x, J > with z ∈ / I. Let w ∈< y, J > . If w ∈ I/J, then < y, J >⊆< x, J >. Suppose w ∈ / I. Then L(L(z)∗ , L(w)∗ ) * I. Since L(L(z)∗ , L(x)∗ ) ⊆ J, we have L(L(z)∗ , L(x)∗ , L(w)∗ ) ⊆ J. By Theorem 2.4, we have L(L(z)∗ , L(x)∗ ) ⊆ J and L(L(x)∗ , L(w)∗ ) ⊆ J, which imply w ∈< x, J >.  Theorem 2.9. Let J ̸= {0} be a semi-prime ideal of P such that J ̸= SP (J) = I1 ∩ I2 , where I1 , I2 are distinct strongly prime ideals of P that are minimal over J and J satisfies (*) condition. Then for each x ∈ SP (J)\J and SP (J) ⊆< x, J >, we have < x, J > is a strongly prime of P containing I1 and I2 . Furthermore either < y, J >⊆< x, J > or < x, J >⊆< y, J > for every x, y ∈ SP (J)\J. Proof. Let x ∈ (I1 ∩ I2 )\J. Then by Theorem 2.7, L(I1∗ , I2∗ ) ⊆ J, which implies I1 ⊆< x, J > and I2 ⊆< x, J >. Let L(A∗ , B ∗ ) ⊆< x, J > for some different ideals A, B of P . If A, B ⊆ I1 or A, B ⊆ I2 , then I1 ⊆< x, J > or I2 ⊆< x, J > . If A, B * I1 or A, B * I2 , then L(A∗ , B ∗ ) * J. Since L(A∗ , B ∗ ) ⊆< x, J >, we have L(A∗ , B ∗ , L(x)∗ ) ⊆ J. By Theorem 2.4, we have L(A∗ , L(x)∗ ) ⊆ J and L(B ∗ , L(x)∗ ) ⊆ J. Hence A ⊆< x, J > and B ⊆< x, J >. It follows from Theorem 732 Catherine Grace John and B. Elavarasan 2.8 that either < y, J >⊆< x, J > or < x, J >⊆< y, J > for every x, y ∈ SP (J)\J. /, Following [4] and [5], for an ideal I of P , an element x ∈ P is called prime to I if < x, I >= I. A proper ideal I of P is called primal if the set S(I) = {x ∈ P : x is not prime to I} is an ideal of P and it is called the adjoint set of I. It is clear that for any ideal I of P , I ⊆ S(I) and S(I) is a semi-ideal of P , but S(I) is not necessarily to be an ideal of P as shows in the following example. Example 2.10. Consider P = {0, a, b, c, d} and define the partial relation ≤ on P as follows d b b b b c b a b 0 Then (P, ≤) is a poset and I = {0} is an ideal of P . Here S(I) = {0, a, b, c} is a semi-ideal of P , but not an ideal of P as L(U (b, c)) = {0, a, b, c, d} * S(I).  If I is a proper ideal of P , then I = S(I) ([9], Theorem 20), so every proper prime ideal of P is primal. But Example 2.11 shows that there exists a primal ideal of P which is not necessarily to be prime. Example 2.11. Consider P = {0, a, b, c} and define a relation ≤ on P as follows. c b a b b b b 0 Then (P, ≤) is a poset with a primal ideal I = {0}. Here I is not a prime ideal of P .  Following [7], a proper ideal I of P is called irreducible if for any ideals J and K of P, I = J ∩ K implies J = I or K = I. Lemma 2.12. Every prime ideal of a poset P is irreducible. Strongly Prime Ideals and Primal Ideals in Posets 733 Proof. Let I be a prime ideal of P such that I = J ∩ K for some ideals J and K of P . Then I ⊆ J and I ⊆ K. Suppose I ̸= J and I ̸= K. Then there exist x, y ∈ P such that x ∈ J\I and y ∈ K\I, which imply L(x, y) ⊆ J ∩ K ⊆ I, so either x ∈ I or y ∈ I, a contradiction.  In Example 2.10, I = {0, a} is an irreducible ideal of P , but not primal. So the converse of Lemma is not true in general. But we have the following. Theorem 2.13. Every irreducible semi-prime ideal of P is primal. Proof. Let I be an irreducible semi-prime ideal of P and x1 , x2 ∈ S(I). Then I ⊂< x1 , I > ∩ < x2 , I > and there exists a ∈ (< x1 , I > ∩ < x2 , I >)\I such that L(a, x1 ) ⊆ I and L(a, x2 ) ⊆ I. Since I is semi-prime, we have L(a, U (x1 , x2 )) ⊆ I. So for any t ∈ L(U (x1 , x2 )), we have L(a, t) ⊆ I.  For any ideal I of P , if S(I) is a strongly prime ideal of P , then I is called a S-primal ideal of P . Following [2], for an ideal I and a strongly ∪ prime ideal Q of < s, I >. P , we define IQ = {x ∈ P : L(s, x) ⊆ I for some s ∈ P \Q} = s∈P \Q Lemma 2.14 Let I be a Q-primal ideal of P with (*) condition. Then < x, I >Q =< x, IQ >. Proof. Let y ∈< x, I >Q . Then there exists c ∈ / Q such that L(y, c) ⊆< x, I >, which implies L(x, t) ⊆ I for all t ∈ L(y, c). Since t ∈ L(y), we have L(L(y), x) ⊆ L(t, x) ⊆ I ⊆ IQ . Then y ∈< x, IQ >. Let a ∈< x, IQ >. Then L(x, a) ⊆ IQ , which implies c ∈ / Q such that L(t, c) ⊆ I for all t ∈ L(x, a). Since c ∈ / Q, we have t ∈< c, I >= I. Hence a ∈< x, I >Q .  Theorem 2.15. Let I be a Q1 -primal ideal of P . Then (i) If Q2 is a strongly prime ideal of P containing Q1 , then IQ2 = I. (ii) If Q2 is a strongly prime ideal of P not containing Q1 , then IQ2 ⊃ I (iii) If IQ2 is a Q2 -primal ideal for some strongly prime ideal Q2 containing I and I satisfies (*) condition, then Q2 ⊆ Q1 . Proof. (i) Let x ∈ IQ2 . Then there exists c ∈ / Q2 such that L(x, c) ⊆ I, which implies c ∈ / Q1 with L(x, c) ⊆ I. Since I is Q1 -primal and c ∈ / Q1 , we have c is prime to I. So x ∈< c, I >= I. (ii) Let x ∈ Q1 \Q2 . Then x is not prime to I and for some y ∈ / I, we have L(x, y) ⊆ I. Since x ∈ / Q2 , we have y ∈ IQ2 . But y ∈ / I. Hence I ⊂ IQ2 . (iii) Suppose Q2 is not a subset of Q1 . Then there exists q ∈ Q2 \Q1 such that L(q, x) ⊆ IQ2 for some x ∈ P \IQ2 , which implies c ∈ P \Q2 such that L(t, c) ⊆ I for all t ∈ L(q, x). Since t ∈ L(x), we have L(L(x), c) ⊆ L(t, c) ⊆ I. However c ∈ / Q2 , so that x ∈ IQ2 , a contradiction.  We now present more properties of primal ideals in a poset by using the following definition. For an ideal I of a poset P , the notation C(I) = {x ∈ P \I : L(x, y) ⊆ I for some y ∈ P \I}. If I is strongly prime, then C(I) = ∅. 734 Catherine Grace John and B. Elavarasan Lemma 2.16. Let I be a proper ideal of P . Then the following hold. (i) I ⊆ S, where S is the adjoint set of I. (ii) C(I) = S\I. Proof. (i) Let x ∈ I. Then L(x, y) ⊆ I with y ∈ / I, which implies x is not prime to I. So x ∈ S. (ii) Let r ∈ C(I). Then r ∈ / I and L(x, r) ⊆ I for some x ∈ / I, which imply r ∈ S. Since r ∈ / I, we have C(I) ⊆ S\I. Conversely, let a ∈ S\I. Then there exists y ∈ /I such that L(a, y) ⊆ I. So a ∈ C(I). Hence C(I) = S\I.  Theorem 2.17. Let I and J be proper ideals of P with I ⊆ J. Then I is a J-primal ideal of P if and only if C(I) = J\I. Proof. Let I be a J-primal ideal of P . Then by Lemma , we have C(I) = S\I = J\I. Let C(I) = J\I. Then it is enough to prove that J is exactly the set of elements that are not prime to I. Let c ∈ J. If c ∈ I, then < c, I >= P ̸= I. So c is not prime to I. If c ∈ J\I = C(I), then there exists z ∈ / I such that L(z, c) ⊆ I. It gives c is not prime to I. Suppose x ∈ / J and x is not prime to I. Then there exists t∈ / I such that L(x, t) ⊆ I, which implies x ∈ C(I) = J\I, a contradiction. We now prove that J is strongly prime. It is enough to prove that S(J) = J. Clearly J ⊆ S(J). Let t ∈ S(J). If t ∈ I, then t ∈ S(J). If t ∈ / I, then there exists s ∈ /I such that L(s, t) ⊆ I, which implies s ∈ C(I) = J\I ⊆ J. Hence I is a J-primal ideal of P .  Corollary 2.18. Let I be an ideal of P . Then I is a primal ideal of P if and only if C(I) ∪ I is an ideal (prime ideal) of P . Corollary 2.19. Let I and J be Q-primal ideals of P . Then C(I) = C(J) if and only if I = J. Lemma 2.20. Let I be an ideal of P with (*) condition. Then I is strongly semi prime of P if and only if < t, I > is a strongly semi-prime ideal of P for any t ∈ P . Proof. It follows directly from Theorem 2.7 of [3].  Theorem 2.21. Let J ̸= {0} be a strongly semi-prime ideal of P such that J ̸= SP (J) ⊆< x, J > for x ∈ SP (J)\J ∪and J satisfies (*) condition. Then J is a < x, J >. Q-primal ideal of P , where Q = x∈SP (J)\J Proof It is clear that J ⊆ Q. We now prove that all elements of Q are not prime to J. Let a, b ∈ P \J such that L(a, b) ⊆ J. It is enough to prove a, b ∈< t, J > for some t ∈ SP (J)\J. By Theorem 2.7, we have SP (J) = I is a strongly prime ideal of P or SP (J) = I1 ∩ I2 , where I1 and I2 are the only distinct strongly prime ideals of P that are minimal over J. If SP (J) = I is a strongly prime ideal of P , then either a ∈ I\J or b ∈ I\J. By Theorem 2.8, we have < a, J >⊆< b, J > or < b, J >⊆< a, J >, which implies Strongly Prime Ideals and Primal Ideals in Posets 735 a, b ∈< a, J > or a, b ∈< b, J > . Then D = {< t, J >: t ∈ SP (J)\J} is a linearly ordered set of ideals and by Lemma , they are strongly semi-prime ideals. Following Theorem 2.8 of [3] and by Zorn’s lemma, there exists a strongly prime ∪ < x, J > of P . ideal Q = x∈SP (J)\J If SP (J) = I1 ∩ I2 , where I1 and I2 are the only distinct strongly prime ideals of P that are minimal over J, then either a ∈ SP (J)\I or a ∈ I1 \I2 and b ∈ I2 \I1 . If a ∈ SP (J)\J, then a, b ∈< a, J >. Suppose that a ∈ I1 \I2 and b ∈ I2 \I1 . Since J ̸= SP (J), there exists d ∈ SP (J)\J. By Theorem 2.9, I1∗ ⊆< d, J > and I2∗ ⊆< d, J >, which imply a, b ∈< d, J >. Then D = {< d, J >: d ∈ SP (J)\J} is a linearly ordered set of strongly semi-prime ideals. By ∪Zorn’s lemma and Theorem < x, J > of P .  2.8 of [3], there exists a strongly prime ideal Q = x∈SP (J)\J Acknowledgments. The authors express their sincere thanks to the referee for his/her valuable comments and suggestions which improve the paper a lot. References [1] J. 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