Int. J. Anal. Appl. (2023), 21:38 Bi-ideals and Weak Bi-ideals of Near Left Almost Rings Thiti Gaketem∗ Fuzzy Algebras and Decision-Making Problems Research Unit, Department of Mathematics School of Science, University of Phayao, Phayao 56000, Thailand ∗Corresponding author: thiti.up.ac.th Abstract. In this paper, we define bi-ideals and weak bi-ideals of nLA-ring. We investigate the prop- erties of bi-ideals and weak bi-ideals of nLA-ring. 1. Introduction M.A. Kazim and MD. Naseeruddin defined LA-semigroup as the following; a groupoid S is called a left almost semigroup, abbreviated as LA-semigroup if (ab)c =(cb)a, ∀a,b,c ∈S M.A. Kazim and MD. Naseeruddin [1, Proposition 2.1] asserted that, in every LA-semigroups G a medial law hold (a ·b) · (c ·d)= (a ·c) · (b ·d), ∀a,b,c,d ∈G. Q. Mushtaq and M. Khan [3, p.322] asserted that, in every LA-semigroups G with left identity (a ·b) · (c ·d)= (d ·b) · (c ·a), ∀a,b,c,d ∈G. Further M. Khan, Faisal, and V. Amjid [2], asserted that, if a LA-semigroup G with left identity the following law holds a · (b ·c)= b · (a ·c), ∀a,b,c ∈G. M. Sarwar (Kamran) [5] defined LA-group as the following; a groupoid G is called a left almost group, abbreviated as LA-group, if (i) there exists e ∈G such that ea= a for all a∈G, (ii) for every a∈G there exists a′ ∈G such that, a′a= e, (iii) (ab)c =(cb)a for every a,b,c ∈G. Received: Mar. 2, 2023. 2020 Mathematics Subject Classification. 16Y30. Key words and phrases. nLA-ring; bi-ideal; weak bi-ideal. https://doi.org/10.28924/2291-8639-21-2023-38 ISSN: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-38 2 Int. J. Anal. Appl. (2023), 21:38 Let 〈G, ·〉 be an LA-group and S be a non-empty subset of G and S is itself and LA-group under the binary operation induced by G, the S is called an LA-subgroup of G, then S is called an LA-subgroup of 〈G, ·〉. S.M. Yusuf in [7, p.211] introduces the concept of a left almost ring (LA-ring). That is, a non- empty set R with two binary operations “+” and “·” is called a left almost ring, if 〈R,+〉 is an LA-group, 〈R, ·〉 is an LA-semigroup and distributive laws of “·” over “+” holds. T. Shah and I. Rehman [7, p.211] asserted that a commutative ring 〈R,+, ·〉, we can always obtain an LA-ring 〈R,⊕, ·〉 by defining, for a,b,c ∈ R, a⊕b = b− a and a ·b is same as in the ring. We can not assume the addition to be commutative in an LA-ring. An LA-ring 〈R,+, ·〉 is said to be LA-integral domain if a·b =0, a,b ∈R, then a = 0 or b = 0. Let 〈R,+, ·〉 be an LA-ring and S be a non-empty subset of R and S is itself and LA-ring under the binary operation induced by R, the S is called an LA-subring of R, then S is called an LA-subring of 〈R,+, ·〉. If S is an LA-subring of an LA-ring 〈R,+, ·〉, then S is called a left ideal of R if RS ⊆S. Right and two-sided ideals are defined in the usual manner. By [4] a near-ring is a non-empty set N together with two binary operations “+” and “·” such that 〈N,+〉 is a group (not necessarily abelian), 〈N, ·〉 is a semigroup and one sided distributive (left or right) of “·” over “+” holds. By [8] If a subgroup B of 〈N,+〉 is said to be a bi-ideal of N if BNB∩ (BN)∗B ⊆B. If N has a zero symmetric near-ring a subgroup B of 〈N,+〉 is a bi-ideal if and only if BNB ⊆B. By [9] A subgroup B of 〈N,+〉 is said to be a weak bi-ideal of N if B3 ⊆ B. In this paper we will define bi-ideal of near-ring has a zero symmetric. 2. Near Left Almost Rings T. Shah, F. Rehman and M. Raees [6, pp.1103-1111] introduces the concept of a near left almost ring (nLA-ring). Definition 2.1. [6]. A non-empty set N with two binary operation “+” and “·” is called a near left almost ring (or simply an nLA-ring) if and only if (1) 〈N,+〉 is an LA-group. (2) 〈N, ·〉 is an LA-semigroup. (3) Left distributive property of · over + holds, that is a ·(b+c)= a ·b+a ·c for all a,b,c ∈N. Definition 2.2. [6]. An nLA-ring 〈N,+〉 with left identity 1, such that 1·a= a for all a∈N, is called an nLA-ring with left identity. Definition 2.3. [6]. A non-empty subset S of an nLA-ring N is said to be an nLA-subring if and only if S is itself an nLA-ring under the same binary operations as in N. Int. J. Anal. Appl. (2023), 21:38 3 Definition 2.4. [6]. An nLA-subring I of an nLA-ring N is called a left ideal of N if NI ⊆ I, and I is called a right ideal if for all n,m ∈N and i ∈ I such that (i +n)m−nm ∈ I, and is called two sided ideal or simply ideal if it is both left and right ideal. Definition 2.5. [6]. Let 〈N,+, ·〉 be an nLA-ring. A non-zero element a of N is called a left zero divisor if there exists 0 6= b ∈N such that a ·b =0. Similarly a is a right zero divisor if b ·a=0. If a is both a left and a right zero divisor, then a is called a zero divisor. Definition 2.6. [6]. An nLA-ring 〈D,+, ·〉 with left identity 1, is called an nLA-ring integral domain if it has no left zero divisor. Definition 2.7. [6]. An nLA-ring 〈F,+, ·〉 with left identity 1, is called a near almost field (n-almost field) if and only if each non-zero element of F has inverse under “·” 3. Bi-ideals and Weak Bi-ideals of Near Left Almost Rings Next we defines of a bi-ideals and weak bi-ideals in nLA-ring is defines the same as a bi-ideal and weak bi-ideal in near-ring in [8] and [9]. Definition 3.1. If a LA-subgroup B of 〈N,+〉 is said to be a bi-ideal of N if (BN)B∩(BN)∗B ⊆B. If N has a zero symmetric nLA-ring a LA-subgroup B of 〈N,+〉 is a bi-ideal if and only if (BN)B ⊆B. Lemma 3.1. Let N be a zero symmetric nLA-ring. An LA-subgroup B of N is a bi-ideal if and only if (BN)B ⊆B. Proof. For an LA-subgroup N of 〈N,+〉 if (BN)B ⊆B then B is a bi-ideal of N. Conversely if B is a bi-ideal, we have (BN)B∩(BN)∗B ⊆B. Since N is a zero symmetric nLA-ring, NB ⊆N ∗B, we get (BN)B =(BN)B∩ (BN)B ⊆ (BN)B∩ (BN)∗B ⊆B. Thus (BN)B ⊆B. � Definition 3.2. Let N be an nLA-ring. An LA-subgroup B of 〈N,+〉 is a bi-ideal if (BN)B ⊆B. Theorem 3.1. If B be a bi-ideal of a nLA-ring N and S is an nLA-subring of N. Then B ∩S is a bi-ideal of S. Proof. Since B is a bi-ideal of N we have (BN)B ⊆ B. Assume that C := B∩S. Then (CS)C ⊆ (SS)S ⊆S, since S is a nLA-subring of N and C ⊆S. On the other hand (CS)C ⊆ (BS)B ⊆ (BN)B ⊆B. Hence (CS)C ⊆B∩S =C. Therefore C is a bi-ideal of S. � Theorem 3.2. Let N be an nLA-ring and A,B be bi-ideals of an nLA-ring N. Then A∩B is a bi-ideal of N. 4 Int. J. Anal. Appl. (2023), 21:38 Proof. Since A,B is bi-ideals of an nLA-ring N, we have A∩B is an LA-subgroup 〈N,+〉. Thus [(A ∩ B)N](A ∩ B) ⊆ (AN)(A ∩ B) = [(A ∩ B)N]A ⊆ (AN)A ⊆ A and [(A ∩ B)N](A ∩ B) ⊆ (BN)(A∩B)= [(A∩B)N]B ⊆ (BN)B ⊆B. It following that A∩B is a bi-ideal of N. � Theorem 3.3. The set of all bi-ideal of nLA-ring. Proof. Let {Bi}i∈I be a set of bi-ideal in N and B :=∩i∈IBi. Then (BN)B ⊆ ( ⋂ i∈I BiN) ⋂ i∈I Bi ⊆Bi for every i ∈ I. Thus B is a bi-ideal of N. � Definition 3.3. Let N be an nLA-ring. An element d of N is called distributive if (n+n′)d = nd+n′d for all n,n′ ∈N. Theorem 3.4. Let N be an nLA-ring. If B is a bi-ideal of N then Bn and n′B are bi-ideal of N where n,n′ ∈N and n′ is a distributive element in N. Proof. Since B is a bi-ideal we have Bn and n′B are an LA-subgroup 〈N,+〉. Thus ((Bn)N)(Bn)⊆ (BN)(Bn)= (BN)Bn ⊆Bn. Hence Bn is a bi-ideal of N. Again ((n′B)N)(n′B)⊆ ((n′B)N)B =(n′BN)B ⊆ n′B. Thus n′B are bi-ideal of N. � Corollary 3.1. If B is a bi-ideal of nLA-ring. For b,c ∈ B, if b is a distributive element in N, then bBc is a bi-ideal of N. Proof. Let B be a bi-ideal of nLA-ring and b is a distributive element in N. Then b(n+n′)= bn+dn′ for all n,n′ ∈N. Since B is a bi-ideal we have bBc is an LA-subgroup 〈N,+〉 then ((bBc)N)(bBc)⊆ (BN)B ⊆B. � Definition 3.4. An nLA-ring N is said to be B-simple if it has no proper bi-ideals. Theorem 3.5. Let N be an nLA-ring with more than one element. If N is a near almost field. Then N is a B-simple. Proof. Let N be a near almost field then {0} and N are the only bi-ideals of N. For if 0 6= B is a bi-ideal of N, then for 0 6= b ∈B we get N =Nb and N = bN. Now N = N2 = (bN)(Nb) ⊆ bNb ⊆ B, since B is a bi-ideal of N. Then N = B. Thus N is a B-simple. � The following we defined weak bi-ideal and study properties it. Definition 3.5. An LA-subgroup B of 〈N,+〉 is said to be a weak bi-ideal of N if B3 ⊆B. Int. J. Anal. Appl. (2023), 21:38 5 Theorem 3.6. Every bi-ideal of an nLA-ring is a weak bi-ideal. Proof. Since B3 =(BB)B ⊆ (BN)B ⊆B we have every bi-ideal is a weak bi-ideal. � Theorem 3.7. If B is a weak bi-ideal of a nLA-ring N and S is a nLA-subring of N. Then B∩S is a weak bi-ideal of N. Proof. Assume that C :=B∩S. Then C3 = ((B∩S)(B∩S))(B∩S) = ((B∩S)(B∩S))B∩ ((B∩S)(B∩S))S ⊆ (BB)B∩SSS = B3 ∩SSS ⊆ B3 ∩SS ⊆ B3 ∩S ⊆ B∩S = C. Thus C3 ⊆C. Hence C is a weak bi-ideal of N. � Theorem 3.8. Let N be an nLA-ring. If B is a weak bi-ideal of N then Bn and n′B are bi-ideal of N where n,n′ ∈N and n′ is a distributive element in N Proof. Since B is a weak bi-ideal we have Bn and n′B are an LA-subgroup 〈N,+〉. Thus (Bn)3 =(BnBn)Bn ⊆ (BB)Bn ⊆B3n =Bn. Hence Bn is a weak bi-ideal of N. Again (n′B)3 =(n′Bn′B)n′B ⊆ (n′BB)B = n′B3 ⊆ n′B. Thus n′B is a weak bi-ideal of N. � Corollary 3.2. If B is a weak bi-ideal of nLA-ring. For b,c ∈ B, if b is a distributive element in N, then bBc is a weak bi-ideal of N. Acknowledgements: This research project was supported by the thailand science research and inno- vation fund and the Department of Mathematics, School of Science, University of Phayao, Phayao 56000, Thailand. Conflicts of Interest: The authors declare that there are no conflicts of interest regarding the publi- cation of this paper. 6 Int. J. Anal. Appl. (2023), 21:38 References [1] M.A. Kazim, Md. Naseeruddin, On almost semigroup, Portugaliae Math. 36 (1977), 41-47. http://eudml.org/ doc/115301. [2] M. Khan, Faisal, V. Amjid, On some classes of Abel-Grassmann’s groupoids, (2010). http://arxiv.org/abs/ 1010.5965. [3] Q. Mushtaq, M. Khan, M-system in LA-semigroups, Southeast Asian Bull. Math. 33 (2009), 321-327. [4] G. Pilz, Near rings, North-Holland, Amsterdam, 1977. [5] M. Sarwar (Kamran), Conditions for LA-semigroup to resemble associative structures, PhD thesis, Quaid-i-Azam University, 1993. [6] T. Shah, F. ur Rehman, M. Raees, On near left almost rings, Int. Math. Forum, 6 (2011), 1103-1111. [7] T. Shah, I. Rehman, On LA-rings of finitely nonzero function, Int. J. Contemp. Math. Sci. 5 (2010), 209-222. [8] T.T. Chelvam, N. Ganesan, On bi-ideal of near-ring, Indian J. Pure Appl. 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