Algebra
Volume 2014 (2014), Article ID 948032, 5 pages
http://dx.doi.org/10.1155/2014/948032

Research Article

Right -Weakly Regular -Semirings

R. D. Jagatap

Y. C. College of Science, Karad, India

Received 20 July 2014; Accepted 11 November 2014; Published 15 December 2014

Academic Editor: K. C. Sivakumar

Copyright © 2014 R. D. Jagatap. 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.

The concepts of a -idempotent -semiring, a right -weakly regular -semiring, and a right pure -ideal of a -semiring are introduced. Several characterizations of them are furnished.

1. Introduction

-semiring was introduced by Rao in [1] as a generalization of a ring, a -ring, and a semiring. Ideals in semirings were characterized by Ahsan in [2], Iséki in [3, 4], and Shabir and Iqbal in [5]. Properties of prime and semiprime ideals in -semirings were discussed in detail by Dutta and Sardar [6]. Henriksen in [7] defined more restricted class of ideals in semirings known as -ideals. Some more characterizations of -ideals of semirings were studied by Sen and Adhikari in [8, 9]. -ideal in a -semiring was defined by Rao in [1] and in [6] Dutta and Sardar gave some of its properties. Author studied -ideals and full -ideals of -semirings in [10]. The concept of a bi-ideal of a -semiring was given by author in [11].

In this paper efforts are made to introduce the concepts of a -idempotent -semiring, a right -weakly regular -semiring, and a right pure -ideal of a -semiring. Discuss some characterizations of a -idempotent -semiring, a right -weakly regular -semiring, and a right pure -ideal of a -semiring.

2. Preliminaries

First we recall some definitions of the basic concepts of -semirings that we need in sequel. For this we follow Dutta and Sardar [6].

Definition 1. Let and be two additive commutative semigroups. is called a -semiring if there exists a mapping denoted by , for all and satisfying the following conditions:

(i); (ii); (iii); (iv), for  all and for all .

Definition 2. An element 0 in a -semiring is said to be an absorbing zero if , , for all and .

Definition 3. A nonempty subset of a -semiring is said to be a sub--semiring of if is a subsemigroup of and , for all and .

Definition 4. A nonempty subset of a -semiring is called a left (resp., right) ideal of if is a subsemigroup of and ) for all , and .

Definition 5. If is both left and right ideal of a -semiring , then is known as an ideal of .

Definition 6. A right ideal of a -semiring is said to be a right -ideal if and such that ; then .
Similarly we define a left -ideal of a -semiring . If an ideal is both right and left -ideal, then is known as a -ideal of .

Example 7. Let denote the set of all positive integers with zero. is a semiring and with , forms a -semiring. A subset of is an ideal of but not a -ideal. Since and but .

Example 8. If is the set of all positive integers, then (, max., min.) is a semiring and with , forms a -semiring. is a -ideal for any .

Definition 9. For a nonempty of a -semiring ,

is called -closure of .

Now we give a definition of a bi-ideal.

Definition 10 (see [11]). A nonempty subset of a -semiring is said to be a bi-ideal of if is a sub--semiring of and .

Example 11. Let be the set of natural numbers and . Then both and are additive commutative semigroups. An image of a mapping is denoted by and defined as product of , for all and . Then forms a -semiring. is a bi-ideal of .

Example 12. Consider a -semiring , where denotes the set of natural numbers with zero and . Define = usual matrix product of , and , for .

is a bi-ideal of a -semiring .

Definition 13. An element 1 in a -semiring is said to be an unit element if , for all and for all .

Definition 14. A -semiring is said to be commutative if , for all and for all .

Some basic properties of -closure are given in the following lemma.

Lemma 15. For nonempty subsets and of , we have the following.

(1)If , then .(2) is the smallest (left -ideal, right -ideal) -ideal containing (left -ideal, right -ideal) -ideal of .(3) if and only if is a (left -ideal, right -ideal) -ideal of .(4), where is a (left -ideal, right -ideal) -ideal of .(5), where and are (left -ideals, right -ideals) -ideals of .

Some results from [11] are stated which are useful for further discussion.

Result 1. For each nonempty subset of , the following statements hold.

(i) is a left ideal of .(ii) is a right ideal of .(iii) is an ideal of .

Result 2. For , the following statements hold.

(i) is a left ideal of .(ii) is a right ideal of .(iii) is an ideal of .

Now onwards denotes a -semiring with an absorbing zero and an unit element unless otherwise stated.

3. -Idempotent -Semiring

In this section we introduce and characterize the notion of a -idempotent -semiring.

Definition 16. A subset of a -semiring is said to be -idempotent if .

Definition 17. A -semiring is said to be -idempotent if every -ideal of is -idempotent.

Theorem 18. In the following statements are equivalent.

(1) is -idempotent.(2)For any , .(3)For every , .

Proof. . Suppose that is a -idempotent -semiring. For any , . Then . Hence by assumption .
Therefore .
Hence . Therefore .
. Let and . Hence by assumption we have . Therefore . Thus we get .
. Let be any -ideal of . Then by assumption . As is a -ideal of , . Therefore , which shows that is a -idempotent -semiring.

Definition 19. A sub--semiring of is a -interior ideal of if and if and such that , then .

Theorem 20. If is a -idempotent -semiring, then a subset of is a -ideal if and only if it is a -interior ideal.

Proof. Let be a -idempotent -semiring. As every -ideal is a -interior ideal, one part of theorem holds. Conversely, suppose a subset of is a -interior ideal of . To show is a -ideal of , let and . As is a -idempotent -semiring, (see Theorem 18). Therefore, for any , . Similarly we can show that . Therefore is a -ideal of .

Theorem 21. is -idempotent if and only if , for any -interior ideals and of .

Proof. Suppose a -semiring is -idempotent. Let and be any two -interior ideals of . Then, by Theorem 20, and are two -ideals of . Hence and . Therefore . By assumption . Therefore . Conversely, let be any -ideal of . As every -ideal of is a -interior ideal of , by assumption, . Therefore is a -idempotent -semiring.

4. Right -Weakly Regular -Semiring

Definition 22. A -semiring is said to be right -weakly regular if, for any , .

Theorem 23. In , the following statements are equivalent.

(1) is right -weakly regular.(2), for each right -ideal of .(3), for a right -ideal and a -ideal of .

Proof. . For any right -ideal of , . Hence . For the reverse inclusion, let . As is right -weakly regular, . Thus , for each right -ideal of .
. For any , and is a right -ideal of , then by assumption . Therefore . Hence is right -weakly regular.
. Let be a right -ideal and be a -ideal of . Then is a right -ideal of . By assumption . Consider . Clearly and . Then and , since is a right -ideal and is a two sided -ideal of . Therefore . Hence .
. Let be a right -ideal of and let be two sided ideal generated by . Then . By assumption . Hence . Therefore .

Theorem 24. is right -weakly regular if and only if every right -ideal of is semiprime.

Proof. Suppose is right -weakly regular. Let be a right -ideal of such that , for any right -ideal of . . Then . Therefore . Hence is a semiprime right -ideal of . Conversely, suppose every right -ideal of is semiprime. Let be a right -ideal of . is also a right -ideal of . By assumption is a semiprime right -ideal of . implies . Therefore . Therefore . Hence is right -weakly regular by Theorem 23.

Definition 25 (see [12]). A lattice is said to be Brouwerian if, for any , the set of all satisfying the condition contains the greatest element.
If is the greatest element in this set, then the element is known as the pseudocomplement of relative to and is denoted by .

Thus a lattice is a Brouwerian if exists for all .

Let denote the family of all -ideals of . Then is a partially ordered set. As and , for all and is an indexing set, we have is a complete lattice under and defined by and . Further we have the following.

Theorem 26. If is a right -weakly regular -semiring, then is a Brouwerian lattice.

Proof. Let and be any two -ideals of . Consider the family of -ideals . Then by Zorn’s lemma there exists a maximal element in . Select such that . By Theorem 23, we have . To show that . Let . Then , where , , and . Therefore for some (see Definition 9).
, as and . Hence implies , since is a -ideal. Therefore, by Theorem 23, . But, by the maximality, we have which implies . Hence is Brouwerian.

As satisfies infinite meet distributive property property of lattice, we have the following.

Corollary 27. If is a right -weakly regular -semiring, then is a distributive lattice (see Birkoff [12]).

Theorem 28. If is a right -weakly regular -semiring, then a -ideal of is prime if and only if is irreducible.

Proof. Let be a right -weakly regular -semiring and let be a -ideal of . If is a prime -ideal of , then clearly is an irreducible -ideal. Suppose is an irreducible -ideal of . To show that is a prime -ideal. Let and be any two -ideals of such that . Then, by Theorem 23, we have . Hence . As is a distributive lattice, we have . Therefore is an irreducible -ideal implies or . Then or . Therefore is a prime -ideal of .

As a generalization of a fully prime semiring defined by Shabir and Iqbal in [5], we define a fully -prime -semiring in [10] as follows.

A -semiring is said to be a fully -prime -semiring if each -ideal of is a prime -ideal.

Theorem 29. A -semiring is a fully -prime -semiring if and only if is right -weakly regular and the set of -ideals of is a totally ordered set by the set inclusion.

Proof. Suppose that a -semiring is a fully -prime -semiring. Therefore every -ideal of is a prime -ideal. As every prime -ideal is a semiprime -ideal, we have which is a right -weakly regular -semiring by Theorem 24. For any two -ideals and of , . By assumption is a prime -ideal and hence we have or . But then or . Hence either or . This shows that the set of -ideals of is a totally ordered set by the set inclusion. Conversely, assume is right -weakly regular and the set of -ideals of is a totally ordered set by set inclusion. To show that is a fully -prime -semiring. Let be a -ideal of and , for any -ideals and of . Then . Hence by Theorem 23, we have . By assumption either or . Therefore or . Hence either or . This shows that is a prime -ideal of .
Therefore is a fully -prime -semiring.

Now we define a -bi-ideal of a -semiring.

Definition 30. A nonempty subset of a -semiring is said to be a -bi-ideal of if is a sub--semiring of , , and for and such that ; then .

Theorem 31. is right -weakly regular if and only if , for any -bi-ideal and -ideal of .

Proof. Suppose is a right -weakly regular -semiring. Let be a -bi-ideal and let be a -ideal of . Let . Then . Therefore . Conversely, let be a right -ideal of . Then itself is a -bi-ideal of . Hence by assumption . Therefore . Hence, by Theorem 23, is a right -weakly regular -semiring.

Theorem 32. is right -weakly regular if and only if , for any -bi-ideal , -ideal , and a right -ideal of .

Proof. Suppose is a right -weakly regular -semiring. Let be a -bi-ideal, let be a -ideal, and let be a right -ideal of . Let . Then . Therefore . Conversely, for a right -ideal of , itself is being a -bi-ideal and itself is being a -ideal of . Then by assumption . Therefore . Therefore . Then, by Theorem 23, is right -weakly regular.

5. Right Pure -Ideals

In this section we define a right pure -ideal of a -semiring and furnish some of its characterizations.

Definition 33. A -ideal of a -semiring is said to be a right pure -ideal if, for any , .

Theorem 34. A -ideal of is right pure if and only if , for any right -ideal of .

Proof. Let be a right pure -ideal and let be a right -ideal of . Then clearly . Now let . As is a right pure -ideal, . This gives . By combining both the inclusions we get . Conversely, assume the given statement holds. Let be -ideal of and . denotes a right -ideal generated by and , where denotes the set of nonnegative integers. Then (see Result 2). Therefore is a right pure -ideal of .

Theorem 35. The intersection of right pure -ideals of is a right pure -ideal of .

Proof. Let and be right pure -ideals of . Then for any right -ideal of we have and . We consider . Therefore is a right pure -ideal of .

Characterization of a right -weakly regular -semiring in terms of right pure -ideals is given in the following theorem.

Theorem 36. is right -weakly regular if and only if any -ideal of is right pure.

Proof. Suppose that is a right -weakly regular -semiring. Let be a -ideal and let be a right -ideal of . Then, by Theorem 23, . Hence, by Theorem 34, any -ideal of is right pure. Conversely, suppose that any -ideal of is right pure. Then, from Theorems 34 and 23, we get that is a right -weakly regular -semiring.

6. Space of Prime -Ideals

Let be a -semiring and let be the set of all prime -ideals of . For each -ideal of define and

Theorem 37. If is a right -weakly regular -semiring, then forms a topology on the set . There is an isomorphism between lattice of -ideals and (lattice of open subsets of ).

Proof. As is a -ideal of and each -ideal of contains , we have the following:
. Therefore . As itself is a -ideal, imply . Now let for ; is an indexing set, and is a -ideal of . Therefore . As , is a -ideal of . Therefore . Further let , .
Let ; is a prime -ideal of . Hence and . Suppose that . In a right weakly regular -semiring, prime -ideals and strongly irreducible -ideals coincide. Therefore is a strongly irreducible -ideal of . As is a strongly irreducible -ideal of , or , which is a contradiction to and . Hence implies . Therefore . Now let . Then implies and . Therefore and imply . Thus . Hence . Thus forms a topology on the set .
Now we define a function by , for all . Let . Consider  .
Consider . Therefore is a lattice homomorphism. Now consider . Then .
Suppose that . Then there exists such that . As is a proper -ideal of , there exists an irreducible -ideal of such that and (see Theorem  6 in [10]). Hence . As is a right -weakly regular -semiring, is a prime -ideal of by Theorem 28. Therefore implies , which is a contradiction. Therefore . Thus implies and hence is one-one. As is onto the result follows.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The author is thankful for the learned referee for his valuable suggestions.

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