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Research ArticleRight ð-Weakly Regular Î-Semirings
R. D. Jagatap
Y. C. College of Science, Karad, India
Correspondence should be addressed to R. D. Jagatap; [email protected]
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, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The concepts of a ð-idempotent Î-semiring, a right ð-weakly regular Î-semiring, and a right pure ð-ideal of a Î-semiring areintroduced. Several characterizations of them are furnished.
1. Introduction
Î-semiring was introduced by Rao in [1] as a generalizationof a ring, a Î-ring, and a semiring. Ideals in semirings werecharacterized by Ahsan in [2], Iseki in [3, 4], and Shabir andIqbal in [5]. Properties of prime and semiprime ideals in Î-semirings were discussed in detail by Dutta and Sardar [6].Henriksen in [7] defined more restricted class of ideals insemirings known as ð-ideals. Some more characterizationsof ð-ideals of semirings were studied by Sen and Adhikari in[8, 9]. ð-ideal in a Î-semiring was defined by Rao in [1] andin [6] Dutta and Sardar gave some of its properties. Authorstudied ð-ideals and full ð-ideals of Î-semirings in [10]. Theconcept of a bi-ideal of a Î-semiring was given by author in[11].
In this paper efforts are made to introduce the conceptsof a ð-idempotent Î-semiring, a right ð-weakly regular Î-semiring, and a right pure ð-ideal of a Î-semiring. Discusssome characterizations of a ð-idempotent Î-semiring, a rightð-weakly regular Î-semiring, and a right pure ð-ideal of a Î-semiring.
2. Preliminaries
First we recall some definitions of the basic concepts of Î-semirings that we need in sequel. For this we follow Duttaand Sardar [6].
Definition 1. Let ð and Î be two additive commutativesemigroups. ð is called a Î-semiring if there exists a mapping
ð à Πà ð â ð denoted by ððŒð, for all ð, ð â ð and ðŒ â Î
satisfying the following conditions:
(i) ððŒ(ð + ð) = (ððŒð) + (ððŒð);
(ii) (ð + ð)ðŒð = (ððŒð) + (ððŒð);
(iii) ð(ðŒ + ðœ)ð = (ððŒð) + (ððœð);
(iv) ððŒ(ððœð) = (ððŒð)ðœð, for all ð, ð, ð â ð and for all ðŒ, ðœ âÎ.
Definition 2. An element 0 in a Î-semiring ð is said to be anabsorbing zero if 0ðŒð = 0 = ððŒ0, ð + 0 = 0 + ð = ð, for allð â ð and ðŒ â Î.
Definition 3. Anonempty subsetð of a Î-semiring ð is said tobe a sub-Î-semiring of ð if (ð, +) is a subsemigroup of (ð, +)and ððŒð â ð, for all ð, ð â ð and ðŒ â Î.
Definition 4. A nonempty subset ð of a Î-semiring ð is calleda left (resp., right) ideal of ð if ð is a subsemigroup of (ð, +)and ð¥ðŒð â ð (resp., ððŒð¥ â ð) for all ð â ð, ð¥ â ð and ðŒ â Î.
Definition 5. If ð is both left and right ideal of a Î-semiringð, then ð is known as an ideal of ð.
Definition 6. A right ideal ðŒ of a Î-semiring ð is said to be aright ð-ideal if ð â ðŒ and ð¥ â ð such that ð+ð¥ â ðŒ; then ð¥ â ðŒ.
Similarly we define a left ð-ideal of a Î-semiring ð. If anideal ðŒ is both right and left ð-ideal, then ðŒ is known as a ð-ideal of ð.
Hindawi Publishing CorporationAlgebraVolume 2014, Article ID 948032, 5 pageshttp://dx.doi.org/10.1155/2014/948032
2 Algebra
Example 7. Letð0denote the set of all positive integers with
zero. ð = ð0is a semiring and with Î = ð, ð forms a Î-
semiring. A subset ðŒ = 3ð0\ {3} of ð is an ideal of ð but not a
ð-ideal. Since 6 and 9 = 3 + 6 â ðŒ but 3 ï¿œâ ðŒ.
Example 8. If ð = ð is the set of all positive integers, then(ð, max., min.) is a semiring and with Î = ð, ð forms a Î-semiring. ðŒ
ð= {1, 2, 3, . . . , ð} is a ð-ideal for any ð â ðŒ.
Definition 9. For a nonempty ðŒ of a Î-semiring ð,
ðŒ = {ð â ð | ð + ð¥ â ðŒ, for some ð¥ â ðŒ} . (1)
ðŒ is called ð-closure of ðŒ.
Now we give a definition of a bi-ideal.
Definition 10 (see [11]). A nonempty subset ðµ of a Î-semiringð is said to be a bi-ideal of ð if ðµ is a sub-Î-semiring of ð andðµÎðÎðµ â ðµ.
Example 11. Letð be the set of natural numbers and Î = 2ð.Then bothð and Î are additive commutative semigroups. Animage of a mappingð à Πà ð â ð is denoted by ððŒð anddefined as ððŒð = product of ð, ðŒ, ð, for all ð, ð â ð and ðŒ â Î.Thenð forms a Î-semiring. ðµ = 4ð is a bi-ideal ofð.
Example 12. Consider a Î-semiring ð = ð2Ã2(ð0), where
ð0denotes the set of natural numbers with zero and Î =
ð. Define ðŽðŒðµ = usual matrix product of ðŽ, ðŒ and ðµ, forðŽ, ðŒ, ðµ â ð.
ð = {(0 ð¥
0 ðŠ) | ð¥, ðŠ â ð
0} (2)
is a bi-ideal of a Î-semiring ð.
Definition 13. An element 1 in a Î-semiring ð is said to be anunit element if 1ðŒð = ð = ððŒ1, for all ð â ð and for all ðŒ â Î.
Definition 14. A Î-semiring ð is said to be commutative ifððŒð = ððŒð, for all ð, ð â ð and for all ðŒ â Î.
Some basic properties of ð-closure are given in thefollowing lemma.
Lemma 15. For nonempty subsets ðŽ and ðµ of ð, we have thefollowing.
(1) If ðŽ â ðµ, then ðŽ â ðµ.
(2) ðŽ is the smallest (left ð-ideal, right ð-ideal) ð-idealcontaining (left ð-ideal, right ð-ideal) ð-ideal ðŽ of ð.
(3) ðŽ = ðŽ if and only if ðŽ is a (left ð-ideal, right ð-ideal)ð-ideal of ð.
(4) ðŽ = ðŽ, where ðŽ is a (left ð-ideal, right ð-ideal) ð-idealof ð.
(5) ðŽÎðµ = ðŽÎðµ, where ðŽ and ðµ are (left ð-ideals, rightð-ideals) ð-ideals of ð.
Some results from [11] are stated which are useful forfurther discussion.
Result 1. For each nonempty subset ð of ð, the followingstatements hold.
(i) ðÎð is a left ideal of ð.(ii) ðÎð is a right ideal of ð.(iii) ðÎðÎð is an ideal of ð.
Result 2. For ð â ð, the following statements hold.(i) ðÎð is a left ideal of ð.(ii) ðÎð is a right ideal of ð.(iii) ðÎðÎð is an ideal of ð.
Now onwards ð denotes a Î-semiring with an absorbingzero and an unit element unless otherwise stated.
3. ð-Idempotent Î-Semiring
In this section we introduce and characterize the notion of að-idempotent Î-semiring.
Definition 16. A subset ðŒ of a Î-semiring ð is said to be ð-idempotent if ðŒ = ðŒÎðŒ.
Definition 17. A Î-semiring ð is said to be ð-idempotent ifevery ð-ideal of ð is ð-idempotent.
Theorem 18. In ð the following statements are equivalent.(1) ð is ð-idempotent.(2) For any ð â ð, ð â ðÎðÎðÎðÎð.(3) For every ðŽ â ð, ðŽ â ðÎðŽÎðÎðŽÎð.
Proof. (1) â (2). Suppose that ð is a ð-idempotent Î-semiring. For any ð â ð, (ð) = ðÎð + ðÎð + ðÎðÎð + ð
0ð.
Then ð â (ð) = ðÎð + ðÎð + ðÎðÎð + ð0ð.
Hence by assumption (ð) = (ð)Î(ð) =
(ðÎð + ðÎð + ðÎðÎð + ð0ð)Î(ðÎð + ðÎð + ðÎðÎð + ð
0ð).
Therefore (ð) = (ðÎð + ðÎð + ðÎðÎð + ð0ð)Î
(ðÎð + ðÎð + ðÎðÎð + ð0ð).
Hence (ð) â ðÎðÎðÎðÎð. Therefore ð â ðÎðÎðÎðÎð.(2) â (3). LetðŽ â ð and ð â ðŽ. Hence by assumption we
have ð â ðÎðÎðÎðÎð. Therefore ð â ðÎðŽÎðÎðŽÎð. Thus we getðŽ â ðÎðŽÎðÎðŽÎð.
(3) â (1). Let ðŽ be any ð-ideal of ð. Then by assumptionðŽ â ðÎðŽÎðÎðŽÎð â ðŽÎðÎðŽ â ðŽÎðŽ. As ðŽ is a ð-ideal of ð,ðŽÎðŽ â ðŽ. Therefore ðŽÎðŽ = ðŽ, which shows that ð is a ð-idempotent Î-semiring.
Definition 19. A sub-Î-semiring ðŒ of ð is a ð-interior ideal ofð if ðÎðŒÎð â ðŒ and if ð â ðŒ and ð¥ â ð such that ð + ð¥ â ðŒ, thenð¥ â ðŒ.
Theorem 20. If ð is a ð-idempotent Î-semiring, then a subsetof ð is a ð-ideal if and only if it is a ð-interior ideal.
Algebra 3
Proof. Let ð be a ð-idempotent Î-semiring. As every ð-idealis a ð-interior ideal, one part of theorem holds. Conversely,suppose a subset ðŒ of ð is a ð-interior ideal of ð. To show ðŒ isa ð-ideal of ð, let ð¥ â ðŒ and ð¡ â ð. As ð is a ð-idempotentÎ-semiring, ð¥ â ðÎð¥ÎðÎð¥Îð (see Theorem 18). Therefore,for any ðŒ â Î, ð¥ðŒð¡ â ðÎð¥ÎðÎð¥ÎðÎð â ðÎð¥ÎðÎð¥ÎðÎð â
ðÎðŒÎðÎðŒÎðÎð â ðÎðŒÎð â ðŒ. Similarly we can show thatð¡ðŒð¥ â ðŒ. Therefore ðŒ is a ð-ideal of ð.
Theorem 21. ð is ð-idempotent if and only ifðŽÎðµ = ðŽâ©ðµ, forany ð-interior ideals ðŽ and ðµ of ð.
Proof. Suppose a Î-semiring ð is ð-idempotent. Let ðŽ and ðµbe any two ð-interior ideals of ð. Then, by Theorem 20, ðŽand ðµ are two ð-ideals of ð. Hence ðŽÎðµ â ðŽ and ðŽÎðµ â ðµ.Therefore ðŽÎðµ â ðŽ â© ðµ. By assumption ðŽ â© ðµ = (ðŽ â© ðµ)
2=
(ðŽ â© ðµ)Î(ðŽ â© ðµ) â ðŽÎðµ.ThereforeðŽÎðµ = ðŽâ©ðµ. Conversely,let ðŽ be any ð-ideal of ð. As every ð-ideal of ð is a ð-interiorideal of ð, by assumption, ðŽÎðŽ = ðŽ â©ðŽ = ðŽ. Therefore ð is að-idempotent Î-semiring.
4. Right ð-Weakly Regular Î-Semiring
Definition 22. A Î-semiring ð is said to be right ð-weaklyregular if, for any ð â ð, ð â (ðÎð)2.
Theorem 23. In ð, the following statements are equivalent.
(1) ð is right ð-weakly regular.
(2) ð 2 = ð , for each right ð-ideal ð of ð.
(3) ð â© ðŒ = ð ÎðŒ, for a right ð-ideal ð and a ð-ideal ðŒ of ð.
Proof. (1) â (2). For any right ð-ideal ð of ð, ð 2 = ð Îð â
ð Îð â ð . Hence ð 2 = ð Îð â ð . For the reverse inclusion,let ð â ð . As ð is right ð-weakly regular, ð â (ðÎð)
2=
(ðÎð)Î(ðÎð) â (ð Îð)Î(ð Îð) â ð Îð = ð 2. Thus ð 2 = ð ,for each right ð-ideal ð of ð.
(2) â (1). For any ð â ð, ð â ðÎð â ðÎð and ðÎð is a rightð-ideal of ð, then by assumption (ðÎð)2 = (ðÎð). Thereforeð â (ðÎð)
2. Hence ð is right ð-weakly regular.(2) â (3). Let ð be a right ð-ideal and ðŒ be a ð-ideal of ð.
Then ð â© ðŒ is a right ð-ideal of ð. By assumption (ð â© ðŒ)2 =ð â© ðŒ. Consider ð â© ðŒ = (ð â© ðŒ)2 = (ð â© ðŒ)Î(ð â© ðŒ) â ð ÎðŒ.Clearly ð ÎðŒ â ð and ð ÎðŒ â ðŒ. Then ð ÎðŒ â ð = ð and ð ÎðŒ âðŒ = ðŒ, since ð is a right ð-ideal and ðŒ is a two sided ð-ideal ofð. Therefore ð ÎðŒ â ð â© ðŒ. Hence ð â© ðŒ = ð ÎðŒ.
(3) â (2). Let ð be a right ð-ideal of ð and let (ð ) be twosided ideal generated byð .Then (ð ) = ðÎð Îð. By assumptionð â© (ð ) = ð Î(ð ). Hence ð = (ð Îð)Î(ð Îð) â ð Îð = ð 2.Therefore ð 2 = ð .
Theorem 24. ð is right ð-weakly regular if and only if everyright ð-ideal of ð is semiprime.
Proof. Suppose ð is right ð-weakly regular. Let ð be a rightð-ideal of ð such that ðŽÎðŽ â ð , for any right ð-ideal ðŽ of ð.ðŽ = ðŽÎðŽ. ThenðŽÎðŽ â ð = ð . ThereforeðŽ â ð . Hence ð is asemiprime right ð-ideal of ð. Conversely, suppose every rightð-ideal of ð is semiprime. Let ð be a right ð-ideal of ð. ð Îð isalso a right ð-ideal of ð. By assumption ð Îð is a semiprimeright ð-ideal of ð. ð Îð â ð Îð implies ð â ð Îð . Thereforeð = ð â ð Îð = ð 2. Therefore ð 2 = ð . Hence ð is rightð-weakly regular byTheorem 23.
Definition 25 (see [12]). A latticeL is said to be Brouwerianif, for any ð, ð â L, the set of all ð¥ â L satisfying thecondition ð ⧠𥠆ð contains the greatest element.
If ð is the greatest element in this set, then the element ðis known as the pseudocomplement of ð relative to ð and isdenoted by ð : ð.
Thus a latticeL is a Brouwerian if ð : ð exists for all ð, ð âL.
LetLðdenote the family of all ð-ideals of ð.Then âšL
ð, ââ©
is a partially ordered set. As {0}, ð â Lðand â
ðŒâÎðŒðŒâ Lð,
for all ðŒðŒâ Lðand Î is an indexing set, we have L
ðis a
complete lattice under ⧠and âš defined by ðŒ âš ðœ = ðŒ + ðœ andðŒ ⧠ðœ = ðŒ â© ðœ. Further we have the following.
Theorem 26. If ð is a right ð-weakly regular Î-semiring, thenLðis a Brouwerian lattice.
Proof. Let ðµ and ð¶ be any two ð-ideals of ð. Consider thefamily of ð-ideals ðŸ = {ðŒ â L
ð| ðŒ â© ðµ â ð¶}. Then by
Zornâs lemma there exists a maximal elementð in ðŸ. SelectðŒ â L
ðsuch that ðµâ©ðŒ â ð¶. ByTheorem 23, we have ðµÎðŒ â ð¶.
To show that ðµÎ(ðŒ +ð) â ð¶. Let ð¥ â ðµÎ(ðŒ +ð). Thenð¥ = â
ð
ð=1ðððŒðð¥ð, where ð
ðâ ðµ, ðŒ
ðâ Î, and ð¥
ðâ ðŒ +ð.
Therefore ðð+ð¥ðâ ðŒ+ð for some ð
ðâ ðŒ+ð (seeDefinition 9).
ð¥ = âð
ð=1ðððŒð(ðð+ ð¥ð) â ðµÎ(ðŒ + ð) = (ðµÎðŒ) + (ðµÎð) â
ð¶, as ðµÎðŒ â ð¶ and ðµÎð â ð¶. Hence ðµÎ(ðŒ +ð) â ð¶
implies ðµÎ(ðŒ +ð) â ð¶ = ð¶, since ð¶ is a ð-ideal. Therefore,by Theorem 23, ðµ â© (ðŒ +ð) â ð¶. But, by the maximality,we have ðŒ +ð â ð which implies ðŒ â ð. Hence L
ðis
Brouwerian.
As Lðsatisfies infinite meet distributive property prop-
erty of lattice, we have the following.
Corollary 27. If ð is a right ð-weakly regular Î-semiring, thenLðis a distributive lattice (see Birkoff [12]).
Theorem 28. If ð is a right ð-weakly regular Î-semiring, thena ð-ideal ð of ð is prime if and only if ð is irreducible.
Proof. Let ð be a right ð-weakly regular Î-semiring and let ðbe a ð-ideal of ð. Ifð is a prime ð-ideal of ð, then clearlyð is anirreducible ð-ideal. Supposeð is an irreducible ð-ideal of ð. Toshow thatð is a prime ð-ideal. LetðŽ andðµ be any two ð-idealsof ð such thatðŽÎðµ â ð.Then, byTheorem 23, we haveðŽâ©ðµ â
4 Algebra
ð. Hence (ðŽ â© ðµ) + ð = ð. AsLðis a distributive lattice, we
have (ðŽ + ð) â© (ðµ + ð) = ð. Therefore ð is an irreducible ð-ideal implies ðŽ + ð = ð or ðµ + ð = ð. Then ðŽ â ð or ðµ â ð.Therefore ð is a prime ð-ideal of ð.
As a generalization of a fully prime semiring defined byShabir and Iqbal in [5], we define a fully ð-prime Î-semiringin [10] as follows.
A Î-semiring ð is said to be a fully ð-prime Î-semiring ifeach ð-ideal of ð is a prime ð-ideal.
Theorem 29. A Î-semiring ð is a fully ð-prime Î-semiring ifand only if ð is right ð-weakly regular and the set of ð-ideals ofð is a totally ordered set by the set inclusion.
Proof. Suppose that a Î-semiring ð is a fully ð-prime Î-semiring. Therefore every ð-ideal of ð is a prime ð-ideal. Asevery prime ð-ideal is a semiprime ð-ideal, we have ðwhich isa right ð-weakly regular Î-semiring by Theorem 24. For anytwo ð-ideals ðŽ and ðµ of ð, ðŽÎðµ â ðŽ â© ðµ. By assumptionðŽ â© ðµ is a prime ð-ideal and hence we have ðŽ â ðŽ â© ðµ orðµ â ðŽ â© ðµ. But then ðŽ â© ðµ = ðŽ or ðŽ â© ðµ = ðµ. Hence eitherðŽ â ðµ or ðµ â ðŽ. This shows that the set of ð-ideals of ð is atotally ordered set by the set inclusion. Conversely, assume ðis right ð-weakly regular and the set of ð-ideals of ð is a totallyordered set by set inclusion. To show that ð is a fully ð-primeÎ-semiring. Let ð be a ð-ideal of ð and ðŽÎðµ â ð, for any ð-ideals ðŽ and ðµ of ð. Then ðŽÎðµ â ð. Hence by Theorem 23,we have ðŽ â© ðµ = ðŽÎðµ â ð. By assumption either ðŽ â ðµ orðµ â ðŽ.ThereforeðŽâ©ðµ = ðŽ orðŽâ©ðµ = ðµ. Hence eitherðŽ â ð
or ðµ â ð. This shows that ð is a prime ð-ideal of ð.Therefore ð is a fully ð-prime Î-semiring.
Now we define a ð-bi-ideal of a Î-semiring.
Definition 30. A nonempty subset ðµ of a Î-semiring ð is saidto be a ð-bi-ideal of ð ifðµ is a sub-Î-semiring of ð,ðµÎðÎðµ â ðµ,and for ð â ðµ and ð¥ â ð such that ð + ð¥ â ðµ; then ð¥ â ðµ.
Theorem 31. ð is right ð-weakly regular if and only if ðµ â© ðŒ âðµÎðŒ, for any ð-bi-ideal ðµ and ð-ideal ðŒ of ð.
Proof. Suppose ð is a right ð-weakly regular Î-semiring. Letðµ be a ð-bi-ideal and let ðŒ be a ð-ideal of ð. Let ð â ðµâ©ðŒ.Thenð â (ðÎð)
2= (ðÎð)Î(ðÎð) â ðµÎ(ðÎðŒÎð) â ðµÎðŒ. Therefore
ðµ â© ðŒ â ðµÎðŒ. Conversely, let ð be a right ð-ideal of ð. Thenð itself is a ð-bi-ideal of ð. Hence by assumption ð = ð â©
(ð ) â ð Î(ð ) = ð Î(ðÎð Îð) = (ð Îð)Î(ð Îð) â ð Îð . Thereforeð = ð Îð = ð 2. Hence, by Theorem 23, ð is a right ð-weaklyregular Î-semiring.
Theorem32. ð is right ð-weakly regular if and only ifðµâ©ðŒâ©ð â
ðµÎðŒÎð , for any ð-bi-ideal ðµ, ð-ideal ðŒ, and a right ð-ideal ð ofð.
Proof. Suppose ð is a right ð-weakly regular Î-semiring. Letðµ be a ð-bi-ideal, let ðŒ be a ð-ideal, and let ð be a right ð-idealof ð. Let ð â ðµ â© ðŒ â© ð . Then ð â (ðÎð)
2= (ðÎð)Î(ðÎð) â
(ðÎð)Î(ðÎð)Î(ðÎð)Îð â ðµÎ(ðÎðŒÎð)Î(ð Îð) â ðµÎðŒÎð .Therefore ðµâ©ðŒâ©ð â ðµÎðŒÎð . Conversely, for a right ð-ideal ð of ð, ð itself is being a ð-bi-ideal and ð itself is being a ð-idealof ð.Then by assumptionð â©ðâ©ð â ð ÎðÎð â ð Îð .Thereforeð â ð Îð . Therefore ð = ð Îð = ð 2. Then, by Theorem 23, ðis right ð-weakly regular.
5. Right Pure ð-Ideals
In this section we define a right pure ð-ideal of a Î-semiringð and furnish some of its characterizations.
Definition 33. A ð-ideal ðŒ of a Î-semiring ð is said to be a rightpure ð-ideal if, for any ð¥ â ðŒ, ð¥ â ð¥ÎðŒ.
Theorem 34. A ð-ideal ðŒ of ð is right pure if and only if ð â©ðŒ =ð ÎðŒ, for any right ð-ideal ð of ð.
Proof. Let ðŒ be a right pure ð-ideal and let ð be a right ð-idealof ð. Then clearly ð ÎðŒ â ð â© ðŒ. Now let ð â ð â© ðŒ. As ðŒ is aright pure ð-ideal, ð â ðÎðŒ â ð ÎðŒ. This gives ð â© ðŒ â ð ÎðŒ.By combining both the inclusions we get ð â© ðŒ = ð ÎðŒ. Con-versely, assume the given statement holds. Let ðŒ be ð-ideal ofð and ð â ðŒ. (ð)
ðdenotes a right ð-ideal generated by ð and
(ð)ð= ð0ð + ðÎðŒ, where ð
0denotes the set of nonnegative
integers. Then ð â (ð)ðâ© ðŒ = (ð)
ðÎðŒ = (ð
0ð + ðÎðŒ)ÎðŒ =
(ð0ð + ðÎðŒ)ÎðŒ â ðÎðŒ (see Result 2).Therefore ðŒ is a right pure
ð-ideal of ð.
Theorem 35. The intersection of right pure ð-ideals of ð is aright pure ð-ideal of ð.
Proof. Let ðŽ and ðµ be right pure ð-ideals of ð. Then for anyright ð-idealð of ðwe haveð â©ðŽ = ð ÎðŽ andð â©ðµ = ð Îðµ.Weconsider ð â©(ðŽâ©ðµ) = (ð â©ðŽ)â©ðµ = (ð ÎðŽ)â©ðµ = (ð ÎðŽ)Îðµ =ð Î(ðŽÎðµ) = ð Î(ðŽ â© ðµ).ThereforeðŽâ©ðµ is a right pure ð-idealof ð.
Characterization of a right ð-weakly regular Î-semiringin terms of right pure ð-ideals is given in the followingtheorem.
Theorem 36. ð is right ð-weakly regular if and only if any ð-ideal of ð is right pure.
Proof. Suppose that ð is a right ð-weakly regular Î-semiring.Let ðŒ be a ð-ideal and let ð be a right ð-ideal of ð. Then, byTheorem 23, ð â© ðŒ = ð ÎðŒ. Hence, byTheorem 34, any ð-idealðŒ of ð is right pure. Conversely, suppose that any ð-ideal of ðis right pure. Then, from Theorems 34 and 23, we get that ðis a right ð-weakly regular Î-semiring.
Algebra 5
6. Space of Prime ð-Ideals
Let ð be a Î-semiring and letâðbe the set of all prime ð-ideals
of ð. For each ð-ideal ðŒ of ð define ÎðŒ= {ðœ â â
ð| ðŒ â ðœ} and
ð (âð) = {Î
ðŒ| ðŒ is a ð-ideal of ð} . (3)
Theorem 37. If ð is a right ð-weakly regular Î-semiring, thenð(âð) forms a topology on the set â
ð. There is an isomorphism
between lattice of ð-idealsLðand ð(â
ð) (lattice of open subsets
of âð).
Proof. As {0} is a ð-ideal of ð and each ð-ideal of ð contains{0}, we have the following:
Î{0}
= {ðœ â âð | {0} â ðœ} = Ί. Therefore Ί â ð(â
ð).
As ð itself is a ð-ideal, Îð= {ðœ â â
ð| ð â ðœ} = â
ðimply
âðâ ð(â
ð). Now let Î
ðŒðâ ð(â
ð) for ð â Î; Î is an indexing
set, and ðŒðis a ð-ideal of ð. Therefore Î
ðŒð= {ðœ â â
ð| ðŒð
â ðœ}.As âðÎðŒð= {ðœ â â
ð| âððŒð
â ðœ}, âððŒðis a ð-ideal of ð.
Therefore âðÎðŒð= ÎâððŒðâ ð(â
ð). Further let Î
ðŽ, Îðµâ
ð(âð).Let ðœ â Î
ðŽâÎðµ; ðœ is a prime ð-ideal of ð. Hence ðŽ â ðœ
and ðµ â ðœ. Suppose that ðŽ â© ðµ â ðœ. In a right weakly regularÎ-semiring, prime ð-ideals and strongly irreducible ð-idealscoincide. Therefore ðœ is a strongly irreducible ð-ideal of ð. Asðœ is a strongly irreducible ð-ideal of ð,ðŽ â ðœ orðµ â ðœ, which isa contradiction to ðŽ â ðœ and ðµ â ðœ. Hence ðŽ â© ðµ â ðœ impliesðœ â Î
ðŽâ©ðµ. Therefore Î
ðŽâÎðµâ ÎðŽâ©ðµ
. Now let ðœ â ÎðŽâ©ðµ
.ThenðŽâ©ðµ â ðœ impliesðŽ â ðœ andðµ â ðœ.Therefore ðœ â Î
ðŽand
ðœ â Îðµimply ðœ â Î
ðŽâÎðµ. Thus Î
ðŽâ©ðµâ ÎðŽâÎðµ. Hence
ÎðŽâÎðµ= ÎðŽâ©ðµ
â ð(âð). Thus ð(â
ð ) forms a topology on
the set âð.
Now we define a function ð : Lðâ ð(â
ð) by ð(ðŒ) = Î
ðŒ,
for all ðŒ â Lð. Let ðŒ, ðŸ â L
ð. Consider ð(ðŒ â© ðŸ) = Î
ðŒâ©ðŸ=
ÎðŒâÎðŸ= ð(ðŒ) â© ð(ðŸ).
Consider ð(ðŒ + ðŸ) = ÎðŒ+ðŸ
= ÎðŒâª ÎðŸ= ð(ðŒ) ⪠ð(ðŸ).
Therefore ð is a lattice homomorphism. Now consider ð(ðŒ) =ð(ðŸ). Then Î
ðŒ= ÎðŸ.
Suppose that ðŒ = ðŸ. Then there exists ð â ðŒ such that ð âðŸ. AsðŸ is a proper ð-ideal of ð, there exists an irreducible ð-ideal ðœ of ð such thatðŸ â ðœ and ð â ðœ (seeTheorem 6 in [10]).Hence ðŒ â ðœ. As ð is a right ð-weakly regular Î-semiring, ðœ isa prime ð-ideal of ð by Theorem 28. Therefore ðœ â Î
ðŸ= ÎðŒ
implies ðŒ â ðœ, which is a contradiction.Therefore ðŒ = ðŸ.Thusð(ðŒ) = ð(ðŸ) implies ðŒ = ðŸ and hence ð is one-one. As ð isonto the result follows.
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper.
Acknowledgment
The author is thankful for the learned referee for his valuablesuggestions.
References
[1] M. M. Rao, âÎ-semirings. I,â Southeast Asian MathematicalSociety. Southeast Asian Bulletin of Mathematics, vol. 19, no. 1,pp. 49â54, 1995.
[2] J. Ahsan, âFully idempotent semirings,â Proceedings of the JapanAcademy, Series A, vol. 32, pp. 185â188, 1956.
[3] K. Iseki, âIdeal theory of semiring,â Proceedings of the JapanAcademy, vol. 32, pp. 554â559, 1956.
[4] K. Iseki, âIdeals in semirings,â Proceedings of the JapanAcademy,vol. 34, pp. 29â31, 1958.
[5] M. Shabir and M. S. Iqbal, âOne-sided prime ideals in semir-ings,â Kyungpook Mathematical Journal, vol. 47, no. 4, pp. 473â480, 2007.
[6] T. K. Dutta and S. K. Sardar, âSemiprime ideals and irreducibleideals of Î-semirings,âNovi Sad Journal of Mathematics, vol. 30,no. 1, pp. 97â108, 2000.
[7] M.Henriksen, âIdeals in semiringswith commutative addition,âNotices of the AmericanMathematical Society, vol. 6, p. 321, 1958.
[8] M. K. Sen andM. R. Adhikari, âOn k-ideals of semirings,â Inter-national Journal of Mathematics andMathematical Sciences, vol.15, no. 2, pp. 347â350, 1992.
[9] M. K. Sen and M. R. Adhikari, âOn maximal ð-ideals ofsemirings,â Proceedings of the American Mathematical Society,vol. 118, no. 3, pp. 699â703, 1993.
[10] R. D. Jagatap and Y. S. Pawar, âk-ideals in Î-semirings,â Bulletinof Pure and Applied Mathematics, vol. 6, no. 1, pp. 122â131, 2012.
[11] R. D. Jagatap and Y. S. Pawar, âQuasi-ideals in regular Î-semirings,âBulletin of KeralaMathematics Association, vol. 7, no.2, pp. 51â61, 2010.
[12] G. Birkoff, Lattice Theory, American Mathematical Society,Providence, RI, USA, 1948.
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