PROPERTIES OF STRONGLY PRIME IDEALS AND C -IDEALS IN POSETS

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. In this paper, the concepts of C -ideal are defined and explored the various properties C -ideals in posets. The equivalent conditions for an ideal to be a C -ideal is obtained. Further the relations between strongly prime ideals and C -ideals are discussed.


INTRODUCTION
The concept of z-ideals, which are both algebraic and topological objects played a fundamental role in studying the ideal theory of C(X), the ring of continuous real-valued functions on a completely regular Hausdorff space X.
In 1973, Mason [6] studied z-ideals of commutative rings and he proved that maximal ideals, minimal prime ideals and some other important ideals in commutative rings are z-ideals.
An ideal I of a commutative ring R is called a z-ideal if for each a ∈ I, the intersection of all maximal ideals containing a is contained in I.
The concept of z 0 -ideals is nothing but the generalization of z-ideals. In 2006, K.Samei [7] studied z 0 -ideals and some special commutative ring.

PRELIMINARIES
Throughout this paper (X, ≤) denotes a poset with least element 0. For basic terminology and notation for posets, we refer [5] and [4]. For E ⊆ X, let E l = {x ∈ X : x ≤ e for all e ∈ E} denotes the lower cone of E in X and dually, let E u = {x ∈ X : e ≤ x for all e ∈ E} be the upper cone of E in X.
It is clear that for any subset E of X, we have E ⊆ E ul and E ⊆ E lu . If E ⊆ F, then F l ⊆ E l and F u ⊆ E u . Moreover, E lul = E l and E ulu = E u .
Following [8], a non-empty subset K of X is called semi-ideal if b ∈ K and a ≤ b, then a ∈ K.
An ideal K of X is called semi-prime if (a, b) l ⊆ K and (a, c) l ⊆ K together imply (a, (b, c) u ) l ⊆ K [5]. Given e ∈ X, (e] = L(e) = {x ∈ X : x ≤ e} is the principal ideal of X generated by e.
Following [3], an ideal K of X is called strongly prime if (A * , B * ) l ⊆ K implies that either Following [3], a non-empty sub-set E of X is called m-system if for any e 1 , e 2 ∈ E, there exists r ∈ (e 1 , e 2 ) l such that r ∈ E.

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As a generalization of m-system, we define the notion of strongly m-system as follows, a non-empty subset E of X is called strongly It is clear that an ideal K of X is strongly prime if and only if X\K is a strongly msystem of X. Also every strongly m-system is m-system. But the converse need not be true in general.
For an ideal K of X, a strongly prime ideal Q of X is said to be a minimal strongly prime ideal of K if K ⊆ Q and there exists no strongly prime ideal R of X such that K ⊂ R ⊂ Q.
The set of all strongly prime ideal of X is denoted by Sspec(X) and the set of minimal strongly prime ideals of X is denoted by Smin(X). For any ideal K of X, SP(K) denotes the intersection of all strongly prime ideals of X containing K and SP(X) denotes the intersection all strongly prime ideal of X.
If K = {0}, then we denote SP(K) = SP(X). From [4], the intersection of all prime semi-ideal of X containing K is K for any semi-ideal K of X. But the intersection of all strongly prime ideal of X containing K need not to be Kfor any ideal K of X [3].
Following [3], let J be an ideal of X. An ideal I of X containing J is called z J -ideal if for each a ∈ I, we have X a (J) ⊆ I. Also if I is a z J -ideal of X, then X a (J) = X for any a ∈ I. Clearly every strongly prime ideal of X is z J -ideal. But the converse need not be true always.

MAIN RESULTS
Definition 3.1. Let X be a poset and I be an ideal of X. Then I is called C -ideal of X if ψ(a) ⊆ ψ(b) and a ∈ I implies b ∈ I. Theorem 3.1. Every strongly prime ideal is a C -ideal of X.
Proof: Let S be a strongly prime ideal of X and ψ(a) ⊆ ψ(b), a ∈ S. Since a ∈ S, we have Corollary 3.1. Let I be a maximal strongly semi-prime ideal of X. Then I is C -ideal.
The following example gives the converse of the theorem 3.1 is need not be true in general.
Theorem 3.2. Let S be a unique strongly prime ideal of X and an ideal I of X such that I ⊂ S.
Then I is not a C -ideal of X.
Proof: Let I ⊂ S. Then there exists a x ∈ S\I. Since I is a unique strongly prime ideal of X, we have ψ(x) = ψ(i) for all i ∈ I which gives I is not a C -ideal of X.  (ii) Let ψ(b) ⊆ ψ(a) for a, b ∈ X and S be a strongly prime ideal of X containing (b, c) l .   Then t ∈ a∈Q∈Ψ Q.
Let Q 1 be any strongly prime ideal of X and b ∈ Q 1 .
As a ∈ SP(b), we have a ∈ Q 1 which implies t ∈ Q 1 for all strongly prime ideals containing b. Hence t ∈ SP(b) and SP(a) ⊆ SP(b).
Theorem 3.5. Let J be an ideal of X.Then the following statements are equivalent (i) J is a C -ideal of X.
(iii) SP(a) ⊆ J for all a ∈ J.
(iv) If SP(b) ⊆ SP(a) and a ∈ J implies b ∈ J.
Proof: (i) ⇒ (ii) It is Obvious.  To show that R is a C -ideal of X, let ψ(a) ⊆ ψ(b) and a ∈ R. Then a ∈ J i for some i. Since J i is a C -ideal of X, we have b ∈ J i and b ∈ R. Thus R is a C -ideal of X.
By Zorn's Lemma, there exists a maximal C -ideal K such that K ∩ M = Φ.
Since M is strongly Hence K is a strongly prime ideal of X.
Theorem 3.7. Every C -ideal is a z J -ideal of X.
Proof: Let I be a C -ideal of X. To prove I is z J -ideal, for all a ∈ I and J ⊆ I, let x ∈ X a (J). ψ (a). ⇒ x ∈ ψ(a) for all a ∈ I ⇒ x ∈ SP(a) for all a ∈ I.
By Theorem 3.4, SP(x) ⊆ SP(a) which gives ψ(a) ⊆ ψ(x). Since I is a C -ideal of X and a ∈ I, we have x ∈ I. Hence X a (J) ⊆ I for all a ∈ I. So I is z J -ideal.
(1) In the above Example 3.1, I 1 = {0, 1} is both C -ideal and z J -ideal of X if we take (2) In Example 3.2, I 1 = {0, a, d} is neither C -ideal nor z J -ideal of X.
The converse of the Theorem 3.7 need not be true in general. The below example gives a z Jideal of X which is not C -ideal.  which implies a ∈ Q 1 . So a ∈ J C . Hence J C is a C -ideal of X.
Let R be any C -ideal of X such that R ⊂ J C and x ∈ J C . Then x ∈ R. So J C ⊆ R for all R.
Hence J C is the least C -ideal of X.
Theorem 3.9. Let A and B be any two ideals of X, then the following statements hold exists a C -ideal J 1 such that t / ∈ J 1 and B ⊆ J 1 which gives A ⊆ J 1 . Since t ∈ A C , we have t ∈ J 1 , a contradiction.
(ii) It is trivial.
For any ideal J of X, J C = {K : K is a C -ideal of X and K ⊇ J}. If union of any two ideals of X is again an ideal in X, then we can say that X has ξ property.
Theorem 3.10. Let J be an ideal of X and X has ξ property. Then J C is the greatest C -ideal Containing J.
Proof: Let ψ(b) ⊆ ψ(a) and b ∈ J C . Then there exists a C -ideal Q 1 of X containing J and b ∈ Q 1 which implies a ∈ Q 1 . So a ∈ {K : K is a C -ideal of X and K ⊇ J} = J C . Hence J C is a C -ideal of X.
Let A be any C -ideal of X such that J C ⊂ A and l ∈ A. Then l ∈ {K : K is a C -ideal of X and K ⊇ J}. So x ∈ J C . Hence J C is the greatest C -ideal of X.
Theorem 3.11. Let E and F be any two ideals of X, then the following statements hold (i) if E ⊆ F, then F C ⊆ E C .
Proof: (i) Let E ⊆ F and t ∈ F C = K⊇F K , where K is a C -ideal of X. Then t ∈ K i for some C -ideal K i of X and K i ⊇ F ⊇ E which implies t ∈ E C .
(ii) Clearly, E C ⊆ (E C ) C . Now, let r ∈ (E C ) C = K⊇E C K, where K is a C -ideal containing E C . But E C is the greatest C -ideal containing E C . Therefore r ∈ E C . Hence (E C ) C = E C .
(iii) It is follows from Theorem 3.8 and Theorem 3.10.
(iv) It is trivial.

ACKNOWLEDGMENT
The author expresses heartfelt gratitude to the referee for his/her insightful feedback and recommendations, which significantly enhance the article.