Algebra
Volume 2015 (2015), Article ID 183930, 6 pages
http://dx.doi.org/10.1155/2015/183930

Research Article

On -Absorbing Primary Elements in Lattice Modules

Sachin Ballal and Vilas Kharat

Department of Mathematics, Savitribai Phule Pune University, Pune 411 007, India

Received 18 December 2014; Accepted 31 March 2015

Academic Editor: Andrei V. Kelarev

Copyright © 2015 Sachin Ballal and Vilas Kharat. 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.

Abstract.

Let be a -lattice and let be a lattice module over . Let be a function. A proper element is said to be -absorbing primary if, for and , and together imply or , for some . We study some basic properties of -absorbing primary elements. Also, various generalizations of prime and primary elements in multiplicative lattices and lattice modules as -absorbing elements and -absorbing primary elements are unified.

1. Introduction

A lattice is said to be complete, if, for any subset of , we have . Since every complete lattice is bounded, (or ) denotes the greatest element and (or ) denotes the smallest element of . A complete lattice is said to be a multiplicative lattice, if there is a defined binary operation “” called multiplication on satisfying the following conditions:

(1), for all ,(2), for all ,(3), for all ,(4), for all .

Henceforth, will be simply denoted by .

An element of a multiplicative lattice is said to be prime if implies either or , for . Radical of an element is denoted by and is defined as , for some .

An element of a complete lattice is said to be compact if implies , where . The set of all compact elements of a lattice is denoted by . By a -lattice we mean a multiplicative lattice with a multiplicatively closed set of compact elements which generates under join.

A complete lattice is said to be a lattice module over a multiplicative lattice , if there is a multiplication between elements of and , denoted by for and , which satisfies the following properties:

(1),(2),(3),(4), for all and for all ,

where denotes the greatest element of and denotes the smallest element of .

For and , denote . For , and for , . For , for some positive integer and it is also denoted by . For , we define . An element is said to be weak join principal if it satisfies the following identity for all .

A lattice module over a multiplicative lattice is called a multiplication lattice module if for there exists an element such that .

An element in is said to be prime if implies or , that is, for every and .

An element is called compact if implies for some . If each element of is a join of principal (compact) elements of , then is called principally generated lattice (compactly generated lattice).

Çallıalp et al. [1] studied the concepts of weakly prime and almost prime elements in multiplicative lattices as extensions of, respectively, weakly prime and almost prime ideals in commutative rings. An element in is said to be weakly prime if implies either or and almost prime if and implies either or for . In [2], the authors generalized these concepts, respectively, to weakly primary and almost primary elements in multiplicative lattices. A proper element is said to be weakly primary if implies either or and almost primary if and implies either or for .

The concept of prime elements in multiplicative lattices is further generalised to 2-absorbing and weakly 2-absorbing elements in multiplicative lattices by Jayaram et al. [3]. An element in is said to be 2-absorbing if implies either or or and is said to be weakly 2-absorbing element if implies either or or , for . Joshi and Ballal [4] defined the concept of -prime elements in multiplicative lattices as a generalization of prime elements. An element of a multiplicative lattice is said to be an -prime if we can express it as meet of at most primes, where is a positive integer.

In [5], the authors defined the concepts of -absorbing, weakly -absorbing, and -almost -absorbing elements in multiplicative lattices as generalizations of, respectively, 2-absorbing, weakly 2-absorbing, and almost prime elements, where . An element of a multiplicative lattice is called -absorbing if implies and called weakly -absorbing if, for , implies , , and . An element of a multiplicative lattice is called -almost -absorbing if and together imply for , .

Manjarekar and Bingi [6] unified the theory of generalizations of prime and primary elements in multiplicative lattice as -prime and -primary elements. Let be a function. A proper element is said to be -prime if and implies either or and it is said to be -primary if and implies either or , for .

As a generalization of primary ideals in commutative rings, Badawi et al. [7] introduced the concept of 2-absorbing primary ideals. In this paper, we extend the various generalizations of prime ideals and primary ideals in commutative rings to lattice modules. Also, we unify various generalizations of prime and primary elements in multiplicative lattices and lattice modules, respectively, as -absorbing elements and -absorbing primary elements.

For basic concepts and terminologies of lattice modules, one may refer to [8–11] and for multiplicative lattices, one may refer to [12–14].

2. -Absorbing Primary Elements

We introduce the concepts of -absorbing elements and -absorbing primary elements in lattice modules which generalizes, respectively, the concepts of prime and primary elements in multiplicative lattices and lattice modules (see [1–3, 5, 6]).

Essentially, we have the following definitions, where is a lattice module over a multiplicative lattice and .

Definition 1. A proper element is said to be -absorbing if implies or for , where and .

Definition 2. Let be a function. A proper element is said to be -absorbing if and together imply or for , where and .

Definition 3. A proper element is said to be -absorbing primary if implies or for , where and .

Definition 4. Let be a function. A proper element is said to be -absorbing primary if and together imply or for , where and .

Let be a lattice module over a multiplicative lattice . For a map , we have the following.

(1) defines weakly -absorbing primary elements of .(2) defines almost -absorbing primary elements of .(3) defines -almost -absorbing primary elements of .(4) defines -absorbing primary elements of .

Remark 5. It follows immediately from the definition that any -absorbing primary element of is a -absorbing primary. However, the converse does not necessarily holds. In fact, we have the following theorem in which the converse is true under certain condition.

Theorem 6. Let be a lattice module over a -lattice . Then every -absorbing primary element with is -absorbing primary.

Proof. Suppose that and , for , . If , then as is -absorbing primary, we have or for some , which concludes that is -absorbing primary.
Now, suppose that . We assume that , for all . If , then there exists such that . Hence, and .
Since is -absorbing primary, or for some and so for some or .
Similarly, we assume that, for , , . Also, we assume that , , because if , then , where and so and .
Now, since is -absorbing primary, or , for some .
Also, implies that there exist with . We have and . Now, is -absorbing primary; therefore, , for some or and consequently is -absorbing primary.

From the above theorem, it follows that if is a -primary element of that is not -absorbing primary, then .

Corollary 7. Let be a lattice module over a -lattice . If is weakly -absorbing primary that is not -absorbing primary, then .

Theorem 8. Let be a multiplication lattice module over -lattice and let . Then the following holds.

(1)Let be two functions with , that is, for each . Then is -absorbing primary if it is -absorbing primary.(2)Consider the following statements.

(a) is -absorbing primary.(b) is weakly -absorbing primary.(c) is -absorbing primary.(d) is -almost -absorbing primary.(e) is almost -absorbing primary.

 Then .(3) is -absorbing primary if and only if it is -almost -absorbing primary for .

Proof. (1) Suppose that is -absorbing primary and also suppose that and for and . Since for each , we have . It follows from the fact , , and is -absorbing primary that or for some and so, is -absorbing primary.
(2) By definition, every -absorbing primary element is weakly -absorbing primary and therefore holds.
Suppose that is weakly -absorbing primary and also suppose that and , for and . Then . By the assumption, or for some and so, is -absorbing primary.
Suppose that is -absorbing primary and also suppose that and for , , and . Then and . Since is -absorbing primary, it follows that or for some and so, is -almost -absorbing primary.
The statement (e) is a particular case of the statement (d) for and therefore holds.
(3) Suppose that is -almost -absorbing primary for and also suppose that and for and . Then and for some . Since is -almost -absorbing primary, we have or for some . Consequently, is -absorbing primary.
The converse follows from .

Corollary 9. Let be a lattice module over a -lattice . If is -absorbing primary, where , then is -absorbing primary.

Proof. Suppose that is -absorbing primary. If is -absorbing primary, then, by Theorem 8, it is -absorbing primary. Now, if is not -absorbing primary, then, by Theorem 6, . Consequently, , for each . And, by Theorem 8(3), is -absorbing primary.

Lemma 10. Let be a multiplicative lattice. If is -absorbing primary with , then is also -absorbing.

Proof. (I) Suppose that and , for . If for , then we have for any positive integer and for .
(II) Now, since , there exists a positive integer such that .
(III) As , we have for a positive integer .
From the assumption and (I), (II), and (III), it follows that . Consequently, and therefore is -absorbing primary.

It is easy to observe that if and are multiplicative lattices then is also a multiplicative lattice with componentwise meet, join, and multiplication.

Also, if and are lattice modules over multiplicative lattices and , respectively, then is a lattice module over with componentwise meet, join, and multiplication given by , where and .

Lemma 11. Let and , where is a lattice module over -lattice , for . Then , for and .

Proof. Let . Now, :

  and ,  and , , .

Lemma 12. Let and , where is a lattice module over -lattice , for . Then , for and .

Proof. Let with . Then , where .
Now, for a positive integer :

  and (by Lemma 11), , , .

Consequently,

Next, let be such that :

  and ;  and ;  and for some with and ;  and for some positive integers and .

Choose . Then and :

 ; ; ; .

Consequently,

From and , result follows.

Theorem 13. Let and , where is a lattice module over -lattice , for , and let be a function. If is a weakly -absorbing primary such that , then is -absorbing primary.

Proof. Let and be such that and .
Since , we have and so . Since is weakly -absorbing primary, we have or , for some . This implies that or for some , by Lemmas 11 and 12. Therefore, is -absorbing primary element of .

Theorem 14. Let be a lattice module over a -lattice and . Then is -absorbing primary if and only if or or for some , for and with .

Proof. Suppose that is -absorbing primary and , for and . Let be such that which essentially implies that .
We have the following two cases.
Case 1. If , then as is -absorbing primary, we have or for some . Therefore, or for some .
Case 2. If , then . So .
Now, from Cases 1 and 2, it follows that or or for some .
Conversely, suppose that and , for and . If , then the result is obvious. So, suppose that .
Now, or or for some . Since , we have . But and so or for some . Consequently, is -absorbing primary.

Theorem 15. Let and , where each is a compactly generated lattice module over a -lattice , for . Let such that , where , , , and is -absorbing primary. Then is a -absorbing primary element of , for each with .

Proof. Let , , and be such that and . Thus , , and . As is -absorbing primary, or . Now, by Lemmas 11 and 12, we have or for and consequently, is -absorbing primary element of , for each .

Corollary 16. Let and , where each is a compactly generated lattice module over a -lattice , for . If is -absorbing primary, where , then is -absorbing primary with .

Proof. We have and the result follows from Theorem 15.

Theorem 17. Let be a lattice module over a -lattice . Suppose that is a weak join principal element with and . Then is -almost -absorbing primary, , if and only if it is -absorbing primary.

Proof. Suppose that is -almost -absorbing primary and , for and . If , then or for some and therefore is -absorbing primary.
Now, suppose that . Since , we have .
If , then it follows from the fact , , and is -almost -absorbing primary that or for some and we are done.
So assume that . Then as , we have . Next, and is weak join principal; together they imply that . Consequently, which implies that is -absorbing primary.
The converse follows from Theorem 8(2).

Note. The results pertaining to -absorbing elements are essentially the corollaries to the respective results of -absorbing primary elements, as such the results of -absorbing elements are the immediate consequences of results of -absorbing primary elements in this paper.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This research work is an outcome of the project supported by Board of College and University Development, Savitribai Phule Pune University, Pune.

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