International Mathematical Forum, 3, 2008, no. 26, Ronnason Chinram

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1 International Mathematical Forum, 3, 2008, no. 26, A Note on Quasi-Ideals in Γ-Semirings 1 Ronnason Chinram Department of Mathematics, Faculty of Science Prince of Songkla University, Hat Yai, Songkhla 90112, Thailand ronnason.c@psu.ac.th Abstract In this paper, some properties of quasi-ideals in Γ-semirings are provided. Mathematics Subject Classification: 20M10, 20M17 Keywords: Γ-semirings, quasi-ideals 1 Introduction and Preliminaries The notions of quasi-ideals have been introduced by O. Steinfeld [12] and [13] for rings and semigroups, respectively (See [14] for general properties of quasi-ideals for rings and semigroups). The notion of a Γ-semigroup have been introduced by M. K. Sen in [9] the year Γ-semigroups are a generalization of semigroups. Many classical notions of semigroup have been extended to Γ- semigroup (see [9], [10], [1], [2], [3], [4], [7]). In [11], M. Shabir, A. Ali and S. Batool have given some properties of quasi-ideals in semirings and in [3], the author have given some properties of quasi-ideals in Γ-semigroups. In this paper, some properties of quasi-ideals in Γ-semirings are provided. 2 Γ-semirings Let (S, +) be a commutative semigroup and Γ be a nonempty set. S is called Γ-semiring if S is a Γ-semigroup (that is S satisfies the identities (aγb)μc = aγ(bμc) for all a, b, c S and γ,μ Γ) and for all a, b, c S and γ Γ, aγ(b + c) =aγb + aγc and (b + c)γa = bγa + cγa. 1 This research was supported by the Commission on Higher Education and a grant of Thailand Research Fund (TRF) - MRG

2 1254 R. Chinram Example 2.1. We have known that (N, +) is a semigroup. Let Γ = {1, 2, 3}. Define a mapping N Γ N N by aγb = a γ b for all a, b N and γ Γ where is the usual multiplication. Then N is a Γ-semiring. Example 2.2. Let S be an arbitrary semiring and Γ be any nonempty set. Define a mapping S Γ S S by aγb = ab for all a, b S and γ Γ. It is easy to see that S is a Γ-semiring. Thus a semiring can be considered to be a Γ-semiring. Since every ring is a semiring, a ring can be considered to be a Γ-semiring. Example 2.3. Let S be a Γ-semiring and α be a fixed element in Γ. We define a b = aαb for all a, b S. We can show that (S, +, ) is a semiring. In this paper, we shall assume that S has a Γ-absorbing zero 0, that is a +0=a =0+a and 0γa =0=aγ0 for all a S and γ Γ. A nonempty subset T of a Γ-semiring S is called a subγ-semiring of S if T is a subsemigroup of (S, +) and aγb T for all a, b T and γ Γ. Example 2.4. Let S be a semiring and T be a subsemiring of S. Then T is a subsemigroup of (S, +) and ab T for all a, b T. Let Γ be any nonempty set. Define a mapping S Γ S S by aγb = ab for all a, b S and γ Γ. By Example 2.2, S is a Γ-semiring. We have for all a, b T, aγb = ab T. Then T can be consider to be a subγ-semiring of a Γ-semiring S. Let R be a ring and T be a subring of R. Similarly, R can be consider to be a Γ-semiring and T can be consider to be a subγ-semiring of a Γ-semiring S. A nonempty subset T of a Γ-semiring S is called a left (resp. right) ideal of S if T is a subsemigroup of (S, +) and xγa T (resp. aγx T ) for all a T,γ Γ and x S. If T is both a left ideal and a right ideal of S, then T is called an ideal of S. It is easy to see that every left ideal, right ideal and ideal of S is a subγ-semiring of S. Example 2.5. Let S be a semiring and T be a left ideal (resp. right ideal, ideal) of S. Similar to Example 2.4, T can be consider to be a left ideal (resp. right ideal, ideal) of a Γ-semiring S. Let R be a ring and T be a left ideal (resp. right ideal, ideal) of R. Similarly, T can be consider to be a left ideal (resp. right ideal, ideal) of a Γ-semiring R. Let X be a nonempty subset of a semiring S. By the term left ideal (X) l (resp. right ideal (X) r, ideal (X) i )ofs generated by X, we mean the smallest left ideal (resp. right ideal, ideal) of S containing X, that is the intersection of all left ideals (resp. right ideals, ideals) of S containing X.

3 A note on quasi-ideals in Γ-semirings 1255 Let A and B be two nonempty subsets of a Γ-semiring S and N be the set of all natural numbers. Let A + B = {a + b a A and b B}, AΓB denotes the set of all finite sums of the form a i γ i b i where a i A, γ i Γ and b i B, NA denotes the set of all finite sums of the form n i a i where n i N and a i B. Theorem 2.1. Let S be a Γ-semiring and X be a nonempty subset of S. Then (i) (X) l = NX + SΓX, (ii) (X) r = NX + XΓS, (iii) (X) i = NX + SΓX + XΓS + SΓXΓS. Proof. (i) Let L = NX + SΓX. Since S has a Γ-absorbing zero, so each x X, x =1x +0 L. Hence X L. Let a, b L. Then a = a 1 + a 2 and b = b 1 + b 2 for some a 1,b 1 NX and a 2,b 2 SΓX. It is easy to prove that a 1 + b 1 NX and a 2 + b 2 SΓX. Thusa + b L. We have SΓL = SΓ(NX + SΓX) =SΓNX + SΓSΓX = NSΓX + SΓSΓX SΓX L. Hence L is a left ideal of S containing X. Next, let K be any left ideal of S containing X. Since X K, NX K and SΓX K. SoL = NX + SΓX K. Therefore L is a smallest left ideal of S containing X. So (X) l = L = NX + SΓX. (ii) and (iii) are similar to (i). Corollary 2.2. If X is a subsemigroup of (S, +), then (X) l = X + SΓX, (X) r = X + XΓS and (X) i = X + SΓX + XΓS + SΓXΓS. Proof. Since X is a subsemigroup of (S, +), NX = X. 3 Quasi-ideals of Γ-semirings Let S be a Γ-semiring. By a quasi-ideal Q we mean a subsemigroup Q of (S, +) such that SΓQ QΓS Q. It is clear that every left ideal and right ideal of a Γ-semiring S is a quasi-ideal of S. Moreover, each quasi-ideal of S is a subγ-semiring of S. In fact, QΓQ SΓQ QΓS Q.

4 1256 R. Chinram Example 3.1. Let S be an arbitrary semiring and Q be a quasi-ideal of S. Then Q is a subsemigroup of (S, +) and SQ QS Q. Let Γ be any nonempty set. Define a mapping S Γ S S by aγb = ab for all a, b S and γ Γ. By Example 2.2, S is a Γ-semiring. We have SΓQ QΓS = SQ QS Q. Then Q can be consider to be a quasi-ideal of a Γ-semiring S. Let R be a ring and Q be a quasi-ideal of R. Similarly, R can be consider to be a Γ-semiring and Q can be consider to be a quasi-ideal of a Γ-semiring R. Let X be a nonempty subset of a Γ-semiring S. The quasi-ideal (X) q of S generated by X, we mean the smallest quasi-ideal ideal of S containing X, that is the intersection of all quasi-ideals of S containing X. Theorem 3.1. Let S be a Γ-semiring and X be a nonempty subset of S. Then (X) q = NX +(SΓX XΓS). Proof. (i) Let Q = NX +(SΓX XΓS). Since S has a Γ-absorbing zero, so each x X, x =1x +0 Q. Hence X Q. Let a, b Q. Then a = a 1 + a 2 and b = b 1 + b 2 for some a 1,b 1 NX and a 2,b 2 SΓX XΓS. It is easy to prove that a 1 + b 1 NX and a 2 + b 2 SΓX XΓS. Thusa + b Q. We have SΓQ QΓS SΓQ = SΓ(NX +(SΓX XΓS)) SΓ(NX + SΓX) SΓX and SΓQ QΓS QΓS =(NX +(SΓX XΓS))ΓS (NX + XΓS)ΓS XΓS. Then SΓQ QΓS SΓX XΓS Q. Hence Q is a quasi-ideal of S containing X. Next, let K be any quasi-ideal of S containing X. Since X K, NX K and SΓX XΓS K. SoQ = NX +(SΓX XΓS) K. Therefore Q is a smallest quasi-ideal of S containing X. So (X) q = Q = NX +(SΓX XΓS). The following theorem is similar to the case of semigroups, Γ-semigroups, rings and semirings. Theorem 3.2. The intersection of a left ideal L and a right ideal R of a Γ-semiring S is a quasi-ideal of S. Proof. It is easy to prove that L R is a subsemigroup of (S, +). We have SΓ(L R) (L R)ΓS SΓL RΓS L R. Hence L R is a quasi-ideal of S.

5 A note on quasi-ideals in Γ-semirings 1257 In case of semigroups and Γ-semigroups it is true that every quasi-ideal can be written as an intersection of a left ideal and a right ideal (See [14] and [3], respectively). However, this result has no analogue for rings (See [14], [5], [8], [6]), by Example 2.5 and Example 3.1, this implies that this result has no analogue for semirings and Γ-semirings. Below we examine Γ-semirings S in which each quasi-ideal is an intersection of a left ideal and a right ideal of S. A left ideal (resp. right ideal, ideal) T of a Γ-semiring S is called a left k-ideal (resp. right k-ideal, k-ideal) of S if a, a + x T, then x T. The following theorem is true. Theorem 3.3. Let Q is a quasi-ideal of a Γ-semiring S. Then (i) If Q SΓQ and SΓQ is a left k-ideal of S, then Q is the intersection of the left ideal Q + SΓQ and the right ideal Q + QΓS. (ii) If Q QΓS and QΓS is a right k-ideal of S, then Q is the intersection of the left ideal Q + SΓQ and the right ideal Q + QΓS. Proof. (i) By Corollary 2.2, we have Q+SΓQ and Q+QΓS is a left ideal and a right ideal of S generated by Q, respectively. Let A =(Q+ SΓQ) (Q+ QΓS). Then Q A. Since Q SΓQ, A = SΓQ (Q + QΓS). To Show A Q, let d SΓQ (Q + QΓS). Then d SΓQ and d = q + n q i γ i a i i=1 where a i A, q, q i Q and γ i Γ. We have q Q SΓQ and q + n i=1 q iγ i a i = d SΓQ. Then n i=1 q iγ i a i SΓQ. Hence n i=1 q iγ i a i SΓQ QΓS, this implies n i=1 q iγ i a i Q. Therefore d = q + n i=1 q iγ i a i Q. (ii) is similar to (i). Let S be a semiring. An element a S is called a left identity (resp. right ideal) of S if x = aγx (resp. x = xγa) for all x S and γ Γ. If a is both a left and right identity, then a is called an identity of S. Corollary 3.4. Let S be a Γ-semiring. The following statements hold. (i) If S has a left identity and every left ideal is a left k-ideal, then every quasi-ideal of S is an intersection of a left ideal and a right ideal of S. (ii) If S has a right identity and every right ideal is a right k-ideal, then every quasi-ideal of S is an intersection of a left ideal and a right ideal of S. Proof. (i) Let Q be a quasi-ideal of S. Since S has a left identity, Q SΓQ. It is easy to prove that SΓQ is a left ideal of S. By assumption, SΓQ is a left k-ideal of S. By Theorem 3.3 (i), Q is an intersection of a left ideal and a right ideal of S. (ii) is similar to (i).

6 1258 R. Chinram The following two theorems are examples of Γ-semirings such that every quasi-ideal of S is the intersection of a left ideal and a right ideal. Theorem 3.5. Let R be a Γ-semiring with identity element 1. Then every quasi-ideal of S is the intersection of a left ideal and a right ideal of S. Proof. Let Q be a quasi-ideal of S. By Corollary 2.2, we have Q + SΓQ and Q + QΓS is a left ideal and a right ideal of S generated by Q, respectively. Let R = Q + QΓS and L = Q + SΓQ. So Q L R. Since S has an identity, Q QΓS and Q SΓQ. Thus R = QΓS and L = SΓQ. Then R L = QΓS SΓQ Q. Therefore Q = R L. An element a of S is regular if there exist x S and α, β Γ such that a = aαxβa. A Γ-semiring S is regular if every element in S is regular. Theorem 3.6. Every quasi-ideal of a regular Γ-semiring S can be written in the form Q = R L for some a right ideal R and a left ideal L of S. Proof. Let S be a regular Γ-semiring and Q be a quasi-ideal of S. By Corollary 2.2, we have Q + SΓQ and Q + QΓS is a left ideal and right ideal of S generated by Q, respectively. Let R = Q + QΓS and L = Q + SΓQ. So Q L R. Let q Q. Since S is regular, there exist x S and α, β Γ such that q = qαxβq. Soq = qα(xβq) QΓS and q =(qαx)βq SΓQ. Thus Q QΓS and Q SΓQ. This implies that L = QΓS and R = SΓQ. Then R L = QΓS SΓQ Q. Therefore Q = R L. ACKNOWLEDGEMENTS. The authors are grateful to the Commission on Higher Education and the Thailand Research Fund (TRF) for grant support. References [1] S. Chattopadhyay, Right inverse Γ-semigroup, Bull. Cal. Math. Soc. 93(2001), [2] S. Chattopadhyay, Right orthodox Γ-semigroup, SEA Bull. Math. 29(2005), [3] R. Chinram, On quasi-gamma-ideals in gamma-semigroups, ScienceAsia 32(2006), [4] R. Chinram and C. Jirojkul, On bi-γ-ideals in Γ-semigroups, Songklanakarin J. Sci. Technol. 29(2007),

7 A note on quasi-ideals in Γ-semirings 1259 [5] R. Chinram and Y. Kemprasit, The intersection property of quasi-ideals in generalized rings of strictly upper triangular matrices, East-West J. Math. 3(2001), [6] R. Chinram and C. Rungsaripipat, The intersection property of quasiideals in generalized rings of strictly upper triangular matrices over mz, Advances in Algebra and Analysis 1(2006), [7] R. Chinram and P. Siammai, On green s relations for Γ-semigroups and reductive Γ-semigroups, Int. J. Alg. 2(2008), [8] Y. Kemprasit and R. Chinram, Generalized rings of linear transformation having the intersection property of quasi-ideals, Vietnam J. Math. 30(2002), [9] M. K. Sen, On Γ-semigroups, Proceeding of International Conference on Algebra and it s Applications, Decker Publication, New York, (1981), 301. [10] M. K. Sen and N. K. Saha, On Γ-semigroup I, Bull. Cal. Math. Soc. 78(1986), [11] M. Shabir, A. Ali and S. Batool, A note on quasi-ideals in semirings, SEA Bull. Math. 27(2004), [12] O. Steinfeld, On ideals quotients and prime ideals, Acta. Math. Acad. Sci. Hung. 4 (1953), [13] O. Steinfeld, Über die quasiideale von halbgruppen, Publ. Math. Debrecen 4(1956), [14] O. Steinfeld, Quasi-ideals in rings and semigroups, Akadémiai Kiadó Budapest, Received: January 30, 2008