CJM Chinese Journal of Mathematics 2314-8071 Hindawi Publishing Corporation 495205 10.1155/2013/495205 495205 Research Article An Element Weakly Primary to Another Element Manjarekar C. S. 1 Kandale U. N. 2 Li Zhao-Yan Ouyang Yao Zhu William 1 Department of Mathematics Shivaji University Kolhapur India unishivaji.ac.in 2 Department of Mathematics Sharad Institute of Technology, College of Engineering Yadrav Ichalkaranji India sitpolytechnic.org 2013 18 11 2013 2013 03 08 2013 25 08 2013 2013 Copyright © 2013 C. S. Manjarekar and U. N. Kandale. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We introduce the concept “An element weakly primary to another element” and using this concept we have generalized some result proved by Manjarekar and Chavan (2004). It is shown that if {bα} is a family of elements weakly primary to a in L, then αbα is weakly primary to a.

1. Introduction

Multiplicative lattice is a complete lattice provided with commutative, associative, and join distributive multiplication for which the largest element 1 acts as a multiplicative identity. A proper element p of L is called prime element if abpap or bp for a,bL and is called primary element if abp implies ap or bnp for some nZ+. An element a of L is called compact if aX, and XL implies the existence of finite number of elements X1,X2,X3,,Xn of L such that aX1X2X3Xn. Throughout this paper, L denotes compactly generated multiplicative lattice with 1 compact and every finite product of compact elements is compact. Let L* be the set of all compact elements in L. Also, (a:b) is the greatest element c in L such that cba. An element aL is join principal if x(y:a)=(xay):a and meet principle if xya=((x:a)y)a, for all x,yL.

An element is principle if it is both join and meet principle. For aL, a={xL*xnaforsomenZ+}. An element aL is called semiprimary if a is primary element. L is said to satisfy the condition (*) if every semiprimary element is primary element.

An element aL is said to be strong join principle element if a is compact and join principle. An element aL is p-primary if a is primary and a=p and aL is semiprime if a=a. An element a of L is called zero divisor if 0bL such that ab=0, and if L has no zero divisor then L will be called lattice domain or simply a domain. L* denotes the set of compact elements of L.

The concept of weakly prime element is studied by Çallialp et al. . The concept of weakly primary element is introduced by Sachin and Vilas . For other definitions and simple properties of multiplicative lattice, one can refer to Dilworth .

Definition 1.

Weakly primary element is defined as follows.

An element qL is said to be a weakly primary element if for a,bL*, 0abq implies aq or bnq for some nZ+.

Example 2.

Lattice of ideals of ring R=Z12,+,· (see Figure 1).

In the lattice of Example 2, an element a is weakly primary element. From Definition 1, it is clear that every weakly prime element is weakly primary element Converse need not be true. Since in Example 2,  a is weakly primary element but it is not weakly prime element. Further, if q is a weakly primary element, then q is a weakly prime element. Because if for compact element x and y such that 0xyq then xnyn=(xy)nq for some nZ+. As q is a weakly primary element, either xnq or (yn)m=ynmq for some nZ+. Consequently, xq or yq. Thus, q is a weakly prime element. This implies that every weakly primary element is a weakly semiprimary element. It need not be true that a is always weakly prime or a is always weakly semiprimary. In Example 2, the least element 0 is not semiprimary as 0=b is not a weakly prime element. The concept of “An element prime to another element” is introduced in . An element bL is prime to an element a if for xL, xba implies xa.

Now, we define the following.

Definition 3.

An element weakly prime to another element is defined as follows.

An element, bL, is called weakly prime to an element aL if for any xL, 0xba implies xa.

In Example 2, the element d is weakly prime to an element c, but d is not weakly prime to any other element of L. This follows directly from the fact that an element y is weakly prime to an element x if and only if x:y=x.

2. An Element Weakly Primary to Another Element

Now we introduce the following main concept which is a generalization of the concept introduced by Manjarekar and Chavan .

Definition 4.

An element b is said to be weakly primary to another element a in L if for xL*, 0xba implies xna for some nZ+.

In Example 2, the element b is weakly primary to a, but note that b is not weakly prime to a. This follows directly from Corollary 9, and note that (a:b)=c=a and a<c. Evidently, if b is weakly prime to a, then b is weakly primary to a in L. Now if p is a weakly prime element and ap, then a is weakly prime to p, and if q is a weakly primary element and a compact element aq, then a is a weakly primary element to q.

Thus, from this, it is clear that elements weakly primary to another element exist in the lattice L. Since L is compactly generated multiplicative lattice with 1 compact, weakly prime element and hence weakly primary element exists in L. Hereafter, L will be a domain. We prove some interesting results including characterizations.

Theorem 5.

No proper nonzero element is weakly prime or weakly primary to itself in L.

Proof.

If a is a proper nonzero element in L and a is weakly primary to a itself, then 0a.1=a implies that 1=a, a contradiction. Therefore, no proper nonzero element is weakly prime or weakly primary to itself in L.

Now we prove some characterizations of an element weakly primary to a.

Theorem 6.

Let aL be a semiprime element. Then 0b is weakly primary to a if and only if b is weakly prime to a.

Proof.

Assume that b is weakly primary to a semiprime element a. Let 0xba for some xL*. Then xna for some nZ+. Consequently, xa. As a is semiprime, xa. Thus, b is weakly prime to a. The converse part is obvious.

Theorem 7.

Let L be a lattice domain. Let a,bL; then 0b is weakly primary to a if and only if (a:b)a.

Proof.

Assume that 0b is weakly primary to a. Let 0xL* such that 0x(a:b); then, 0xba. As b is weakly primary to a,xna for some nZ+. Hence, xa. This shows that (a:b)a.

Conversely, assume that (a:b)a. Let 0xba for some xL*. Then, we have x(a:b)a. This implies that xna for some nZ+. Thus, b is weakly primary to a.

Theorem 8.

Let L be a lattice domain. Let a,bL; then 0b is weakly prime to a if and only if a=(a:b).

Proof.

Assume that 0b is weakly prime to a in L. Let 0xL such that x(a:b). Then 0xba. As b is weakly prime to a, we have xa. This shows that (a:b)a. But a(a:b). Therefore, we get a=(a:b).

Conversely, assume that (a:b)=a. Let, 0xba for xL. Then, we get x(a:b)=a. Thus b is weakly prime to a.

Corollary 9.

Let L be a lattice domain. Let, a,bL. Then, 0b is weakly primary to a but it is nonweakly prime to a if and only if a<(a:b)a.

Proof.

It follows from the fact that a(a:b) and from Theorems 7 and 8.

Corollary 10.

Let a,bL. If a is weakly semiprimary element and ba, then 0b is weakly primary to a.

Proof.

Assume that aL is a weakly semiprimary element and ba. Let xL* such that 0xba. Then 0xba. As a is a weakly prime element and ba, we have xa. This implies that xna for some nZ+. Thus, b is weakly primary to a.

Theorem 11.

Let L be a lattice domain. Let a,bL. Then, 0b is weakly prime to a if and only if b is weakly prime to (a:x) for every 0xL.

Proof.

Assume that b is weakly prime to a in L. Let 0yb(a:x) for some yL. Then, 0xyba. As b is weakly prime to a, xya. Consequently, y(a:x). Thus, b is weakly prime to (a:x) for every 0xL. The converse is obvious, since, if b is weakly prime to (a:x) for every 0xL, b is weakly prime to (a:1)=a.

Theorem 12.

Let a,bL and let L be a lattice domain. If b is weakly primary to a and a is a semiprime element in L, then b is weakly primary to (a:x) for every xL.

Proof.

It follows from Theorems 6 and 11.

Theorem 13.

Let a,bL. Then 0b is weakly primary to a in L if and only if each xa is weakly primary to a.

Proof.

Assume that 0b is weakly primary to a in L. Let xb and yxa for some yL*. Then 0yba. Therefore, by assumption, yna for some nZ+.

This shows that each xb is weakly prime to a. The converse part is obvious.

Theorem 14.

If {bα} is a family of a elements weakly primary to a in L, then αbα is weakly primary to a.

Proof.

It follows from the fact that bααbα and from Theorem 13.

Theorem 15.

Let a,bL. Then, 0b is nonweakly primary to a if and only if each xb is nonweakly primary to a.

Proof.

Assume that b is nonweakly primary to a. Therefore, by Theorem 8, we have a:ba. Let x be an element of L such that xb. Then (a:b)(a:x). This shows that (a:x)a. Thus, again by Theorem 8, each xb is nonweakly primary to a.

This lemma leads us to the following two obvious corollaries.

Corollary 16.

If {bα} is a family of elements nonweakly primary to a in L, then αbα is nonweakly primary to a.

Corollary 17.

If αbα is nonweakly primary to a in L, then each bα is nonweakly primary to a.

Theorem 18.

If y is compact and 0xy is nonweakly primary to a, then either x is nonweakly primary to a or yn is nonweakly primary to a for some nZ+.

Definition 19.

Completely meet semiprimary elements are defined as follows.

An element a is said to be completely meet semiprimary element if a is a completely meet prime element.

Example 20.

Every element x<d is not a completely meet prime element. But note that x=d. Thus, each x<d is a completely meet semiprimary element (see Figure 2).

Result 1.

In any multiplicative lattice S, we have α(a:xα)=a:(αxα) for any a,xαS.

Theorem 21.

Suppose that a is a completely meet semiprimary element of L and let {bα}L. If each 0bα is nonweakly primary to a in L, then αbα is nonweakly primary to a.

Proof.

By Result 1, we have a:(αbα)=α(a:bα). But each 0bα is nonweakly primary to a. Consequently, by Theorem 12, each (a:bα)a. As a is completely meet semiprimary element, α(a:bα)=a:(αbα)a. Therefore, by Theorem 8, αbα is nonweakly primary to a.

Now we construct a new element as follows. Define qa={xLxisnonweaklyprimarytoa}.

Theorem 22.

If a is a completely meet semiprimary element, then qa is weakly primary element and is nonweakly primary to a.

Proof.

By Theorem 21, it follows that qa is nonweakly primary element to a. Let x and y be compact elements such that, 0xyqa. Then by Theorem 15, xy is nonweakly primary to a. Therefore, by Theorem 18, either x is nonweakly primary to a or yn is nonweakly primary to a for some nZ+. Thus, xqa or ynqa for some nZ+. This shows that qa is a weakly primary element.

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