EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 14, No. 3, 2021, 989-1001 ISSN 1307-5543 – ejpam.com Published by New York Business Global On Gamma LA-Rings and Gamma LA-Semirings Waheed Ahmad Khan1,, Abdelghani Taouti2,∗, Azar Salami2, Zahid Hussain1 1 Department of Mathematics, University of Education Lahore Attock Campus, Pakistan 2 ETS-Maths and NS Engineering Division, HCT, University City P. O. Box 7947, Sharjah, United Arab Emirates Abstract. In this note, first we add some new results in Gamma LA-rings and then we initiate the notion of Γ-LA-semirings. Moreover, we introduce and discuss the terms left ideals, right ideals, bi-ideal, quasi ideals, almost prime and weakly almost prime ideals of a Γ-LA-semiring and their characterizations. 2020 Mathematics Subject Classifications: 17D20, 16Y60 Key Words and Phrases: Gamma LA-Rings, Gamma LA-Semirings, quasi ideals of a Gamma- LA-semiring, weakly almost prime ideals of a Gamma-LA-semiring 1. Introduction Γ -ring was introduced by N. Nobusawa in [13] as a generalization of classical rings. Γ -rings have also viewed as the connection with the abelian additive groups of all linear mappings between two finite dimensional spaces over a field. Classical example of Γ -ring presented by Nobusawa was by taking an additive group M consisting of homomorphisms of a module A to a module B and an additive group Γ consisting of homomorphisms of B to A, and aαb the usual composite map, where a, b ∈ M and α ∈ Γ. Barnes introduced radical theory of Γ-rings in [1]. Afterwards, numbers of researchers have been published their research articles on Γ-rings. Similarly, Γ-nearrings were introduced by Satyanarayana in [17]. Booth et al. provided different ways to construct equiprime Γ-nearrings [2]. Γ- semirings were introduced by Rao in [14]. Prime and semi-prime ideals of of Γ-semirings were discussed in [5, 6]. Moreover, Quasi-ideals in Γ-semiring were discussed in [7, 8]. The properties of ideals, prime ideals, semi-prime ideals and their generalization plays a key role in structure of Γ-semirings. However, the properties of an ideal in semirings and Γ-semirings are slightly changed from the properties of the usual ring ideals. Theory of ideals in an ordered Γ-semiring have been introduced in [15]. Similarly, weakly prime and weakly primary ideals in gamma seminearrings have been introduced in [9]. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v14i3.4034 Email addresses: sirwak2003@yahoo.com (W. A. Khan), ganitaouti@yahoo.com.au (Ab. Taouti), asalami@hct.ac.ae (A. Salami), zahidbsc449@gmail.com (Z. Hussain) http://www.ejpam.com 989 c© 2021 EJPAM All rights reserved. W. A. Khan et al. / Eur. J. Pure Appl. Math, 14 (3) (2021), 989-1001 990 A groupoid which satisfies the left invertive law i.e., (xy)z = (zy)x is said to be an LA- groupoid. A groupoid satisfying the medial law i.e., (xy)(zt) = (xz)(yt) holds by groupoid is called medial [3], whereas a groupoid which satisfying the paramedial law i.e., (st)(uv) = (vt)(us) is a paramedial. LA-groupoid S always obeys medial law, whereas paramedial law holds only by LA-groupoid S with left identity e [3]. LA-groupoid S having e as a left identity holds p(qr) = q(pr) [12], a ∈ S is left (right) cancellative if al = am ⇒ l = m (la = ma ⇒ l = m) ∀ l,m ∈ S. If every element is left and right cancellative then S is cancellative and x ∈ S is cancellative if x is left and right cancellative. The notion LA-groupoid to LA-group was extended by Kamran [16]. Similarly, if e is left identity in LA-groupoid (i-e em = m ∀ m ∈ S) and ∀ m ∈ S ∃ m−1 ∈ S such that m−1m = mm−1 = e, then S is called LA-group. LA-semirings are developed by the concepts of LA- semigroup [10, 11]. LA-semiring and certain results on LA-semirings having two variables are described in [4]. A nonempty set R with two binary operation "." and "+" such that (i) (R,+) is LA-group (ii) (R, ·) is LA-groupoid, and non- associative structure w.r.t ′+′ and ′·′ satisfying left and right distributive laws is called LA-ring [20]. LA-ring was further elaborated in [18]. Every x 6= 0 element of left almost ring R has multiplicative inverse x−1 and having left identity e then LA-ring R is called LA-field. LA-ring < R, ⊕, . > can be obtain by defining p ⊕ q = q − p and pq, for p, q, r ∈ R, is similar as in the ring. The addition in LA-ring cannot assume to be commutative. If for p, q ∈ R, pq = 0 implies p = 0 or q = 0 then LA-ring R is called LA-integral domain. If ∅ 6= S ⊆ R and S is LA-ring under binary operation defined in R, then S is LA-subring. If RS ⊆ S, then S is left ideal of R. Similarly we can define right and two-sided ideals. If PQ ∈ A =⇒ P ∈ A or Q ∈ A then ideal A of R is called prime. Left primary and weakly left primary ideals in Γ-LA-rings and their characterizations are presented in [19]. It is well known that an ideal I of a semiring R is called subtractive, if whenever a, a+b ∈ I, bR, we have b ∈ I. Similarly, A left k-ideal I of a semiring S is a left ideal such that if a ∈ and x ∈ S and if either a+ x ∈ I. or x+ a ∈ I, then x ∈ I. In this note, first we add few new theorems and examples in the theory of Γ-LA-rings and then we introduce the notion of Γ-LA-semirings. In due course, we describe c-prime, 3-prime ideals and their relationships among themselves in Γ-LA-ring and Γ-LA-semirings. Finally, we discuss left ideals, right ideals, and some results on bi-ideal, quasi ideals, almost prime and weakly almost prime ideals in Γ-LA-semiring. 2. Main Results and Discussions 2.1. Some applications of prime ideals in Γ-La-ring In this section, we introduce different types of prime ideals in Γ-LA-rings along with their applications. We begin by recalling definition of Γ-LA-ring and then we add few new results and examples in the theory of Γ-LA-ring. Definition 1. [19] Let (R,+) and (Γ ,+) be the two LA-groups and there exists a mapping R×Γ ×R→ R by (a, α, b)→ aαb, for all a, b ∈ R and α ∈ Γ is called a gamma LA-ring, if it satifies the following conditions. W. A. Khan et al. / Eur. J. Pure Appl. Math, 14 (3) (2021), 989-1001 991 1. aα(b+ c) = aαb+ aαc 2. (a+ b)αc = aαc+ bαc 3. a(α+ β)b = aαb+ aβb 4. (aαb)βc = (cαb)βa,∀ a, .b, c ∈ R ,α, β ∈ Γ Example 1. Let R = {a1, a2, a3, a4, a5, a6, a7, a8} be a set with two binary operations ”+” and ”.” given in the Tables set 1 be the LA-ring and Γ = {s1, s2, s3} with binary operation⊕ is LA-group. Tables Set 1 + a1 a2 a3 a4 a5 a6 a7 a8 a1 a1 a2 a3 a4 a5 a6 a7 a8 a2 a3 a1 a4 a2 a7 a5 a8 a6 a3 a2 a4 a1 a3 a6 a8 a5 a7 a4 a4 a3 a2 a1 a8 a7 a6 a5 a5 a5 a6 a7 a8 a1 a2 a3 a4 a6 a7 a5 a8 a6 a3 a1 a4 a2 a7 a6 a8 a5 a7 a2 a4 a1 a3 a8 a8 a7 a6 a5 a4 a3 a2 a1 · a1 a2 a3 a4 a5 a6 a7 a8 a1 a1 a1 a1 a1 a1 a1 a1 a1 a2 a1 a5 a5 a1 a1 a5 a5 a1 a3 a1 a5 a5 a1 a1 a5 a5 a1 a4 a1 a1 a1 a1 a1 a1 a1 a1 a5 a1 a4 a4 a1 a1 a4 a4 a1 a6 a1 a8 a8 a1 a1 a8 a8 a1 a7 a1 a8 a8 a1 a1 a8 a8 a1 a8 a1 a4 a4 a1 a1 a4 a4 a1 ⊕ s1 s2 s3 s1 s1 s2 s3 s2 s3 s1 s2 s3 s2 s3 s1 Then, clearly R is Γ -LA-ring under operation xγy = xy where x, y ∈ R andγ ∈ Γ . Example 2. Let R = {k, l,m, n, o, p, q, r} with two binary operations ” + ” and ”.” given in Tables set 2 be the LA-ring and Γ = {s, t, u, v, w} with binary operation ⊕ is LA-group. Tables Set 2 + k l m n o p q r k m k n l q r o p l n m l k r q p o m k l m n o p q r n l n k m p o r q o q r o p m k n l p r q p o n m l k q o p q r k l m n r p o r q l n k m · k l m n o p q r k k k k k k k k k l k m k m k k m m m k o k o k k o o n k p k p k k p p o k k k k k k k k p k o k o k k o o q k m k m k k m m r k p k p k k p p W. A. Khan et al. / Eur. J. Pure Appl. Math, 14 (3) (2021), 989-1001 992⊕ s t u v w s s t u v w t w s t u v u v w s t u v u v w s t w t u v w s Then R is Γ -LA-ring under operation xγy = x+ γ + y where x, y ∈ R and γ ∈ Γ Definition 2. An ideal I is said to be a c-prime ideal of a Γ-LA-ring R, if x, y ∈ R, γ ∈ Γ and xγy ∈ I =⇒ x ∈ I or y ∈ I. Definition 3. An ideal I is said to be a 3-prime ideal of a Γ-LA-ring R if x, y ∈ R, β ∈ Γ and xβsβy ∈ I for all s ∈ R implies x ∈ I or y ∈ I. We present few relationships among c-prime, 3-prime and prime ideals. Lemma 1. In a Γ-LA-ring R, every c-prime ideal is a 3-prime ideal. Proof. Let I be a c-prime ideal of Γ-LA-ring R. Let x, y ∈ R, β ∈ Γ and xβsβy ∈ I for all s ∈ R. As I is c-prime ideal so x ∈ I or y ∈ I. So I is a 3-prime ideal of R. Lemma 2. In a Γ-LA-ring R, every 3-prime ideal is a prime ideal. Proof. Let I be a 3-prime ideal of Γ-LA-ring R. Let y ∈ I , γ ∈ Γ and xγy ∈ I. As I is 3-prime ideal so x ∈ I or y ∈ I. Clearly, I is prime ideal of R. Lemma 3. Every c-prime ideal in Γ-LA-ring is a prime ideal. Proof. Let I be a c-prime ideal of Γ-LA-ring R, and let y ∈ I, γ ∈ Γ and xγy ∈ I. Since I is c-prime ideal, x ∈ I or y ∈ I. Then I is a prime ideal of R. Theorem 1. Let R be a Γ -LA-ring. Then, I is a 3-prime ideal in R iff R/I is a Γ -LA- integral domain. Proof. (=⇒)Let I be a 3-prime ideal of R, then by Lemma 2, I is a prime ideal. Thus, R/I is a Γ -LA-integral domain. (⇐=) Suppose that R/I is a Γ-LA-integral domain with xβsβy ∈ I for all s ∈ R. Then I + arb = I, so (I + a) (I + b) = I, where a, b ∈ R. Since R/I is Γ -LA-integral domain, we have I + a = I or I + b = I, then aI or bI is in I. Hence I is a 3-prime ideal of R. Theorem 2. Let I be an ideal of a Γ -LA-ring R. Then, I is a c-prime ideal in R iff R/I is a Γ -LA-integral domain. W. A. Khan et al. / Eur. J. Pure Appl. Math, 14 (3) (2021), 989-1001 993 Proof. (⇒) Let I be a c-prime ideal in R. Then I is a prime ideal by Lemma 3. Thus R/I is a Γ -LA-integral domain. (⇐) Assume that R/I is a Γ-LA-integral domain with xβsβy ∈ I for all s ∈ R. Then, I+arb = I so (I + a) (I+b) = I, a, b ∈ R. Since R/I is a Γ -LA-integral domain, I+a = I or I + b = I, then aI or bI is in I. Thus I is a c-prime ideal of R. Here we introduce the notion of weakly prime ideal in Γ-LA-rings. Definition 4. A proper ideal I is said to be a weakly prime ideal of a Γ -LA-ring R if 0 6= AΓB ⊆ I implies either A ⊆ I or B ⊆ I for any ideals A and B of R. Remark 1. Obviously every prime ideal is weakly prime and {0} is always weakly prime ideal. Theorem 3. Let I be a weakly prime ideal of Γ-LA-ring which is not prime. Then I = 0. Proof. Since I is a weakly prime (but not prime), there exist ideals A 6⊆ I and B 6⊆ I but 0 = AΓB ⊆ I. Since I ⊆ A + I and B ⊆ B + I. But, if I2 6= 0, by distributive laws “.” over ” + ” of Γ-LA-ring, we have 0 6= I2 = IΓI ⊆ (A+ I)Γ(B + I) = [(A+ I)ΓB] + [(A+ I)ΓI] = AΓB + IΓB +AΓI + IΓI ⊆ I. Which implies (A + I) ⊆ I and(B + I) ⊆ I, since I is a weakly prime i.e., A ⊆ I or B + I ⊆ I, a contradiction. Hence, I2 = 0. Remark 2. It is clear that if R2 = RΓR = 0 then every ideal of gamma left almost ring is a weakly prime. Theorem 4. In gamma left almost ring R, every ideal is a weakly prime iff AΓB = A, AΓB = B or AΓB = 0, for any ideals A, B of R. Proof. Assume that every ideal in R is a weakly prime ideal. Let A, B be weakly prime ideals of R, then AΓB is a left ideal of R provided that AB 6=R, then by hypothesis, AΓB is weakly prime. We consider two situations, that is AΓB = 0 or AΓB 6= 0. If 0 6= AΓB ⊆ AB, then by Definition 4 we have A ⊆ AΓB or B ⊆ AΓB. Since A and B are ideals of R, we have AΓB ⊆ A and AΓB ⊆ B. Therefore, A = AΓB or B = AΓB. If AΓB = R then A = B = R, whence R2 = R. Conversely, for proper ideal I of R and ideals A and B, suppose that 0 6= AΓB ⊆ I. Then either A = AΓB ⊆ I or B= AΓB ⊆ I. Corollary 1. If every ideal of gamma left almost ring is weakly prime. Then, either A2 = A or A2 = 0 for ideal A. W. A. Khan et al. / Eur. J. Pure Appl. Math, 14 (3) (2021), 989-1001 994 Theorem 5. In Γ -LA-ring every c-prime ideal is a weakly prime ideal. Proof. Let I be c-prime ideal of Γ -LA-ring R, by Lemma 1, I is a prime ideal. So I is a weakly prime ideal of R because every prime ideal is weakly prime ideal. Theorem 6. Every 3-prime ideal in Γ -LA-ring is a weakly prime ideal. Proof. Let I is 3-prime ideal of Γ -LA-ring R, by Lemma 2, we have I is a prime ideal. Since every prime ideal is a weakly prime ideal, we have I a weakly prime ideal of R. 3. Γ-LA-SEMIRINGS In this section, we introduce the notion of Γ-LA-semiring. Furthermore, we introduce prime, weakly prime, subtracted ideals and nilpotent elements in Γ -LA-semiring along with some interesting results. Definition 5. Let (S,+) and (Γ ,+) be the two LA-monoids. Then S is said to be a gamma LA-semiring (or Γ -LA-semiring) if there exists a mapping S×Γ ×S → S written (x, γ, y) by xγy such that the following axioms hold 1. xγ(y + z) = xγy + xγz and (x+ y)γz = xγz + yγz 2. x(γ + β)y = xγy + xβy 3. (xγy)βz = (zγy)βx for all x, y, z ∈ S, γ, β ∈ Γ . In this case we denote Γ -LA-semiring by (S, Γ ). Example 3. Let S = {a, b, c, d, e} with two binary operations ” + ” and ” .” given in the Tables set 3 be the LA-semiring and Γ = {p, q, r, s} with binary operation ” ⊕ ” is LA-monoid. Tables Set 3 + a b c d e a s t u v w b a a d a b c a b b d e d a a b a d e a d e b c · a b c d e a a b c d e b e a b c d c d e a b c d c d e a b e b c d e a ⊕ p q r s p p p p p q p q r s r p s q r s p r s q Then, S is LA- Γ -semiring under operations, (xγy)βz = (zγy)βx and xγy = xy where x, y ∈ S and γ ∈ Γ . Definition 6. Let I be a proper ideal of gamma LA-semiring S and AB ⊆ I such that A ⊆ I or B ⊆ I for any ideals A,B of S, then proper ideal I of a gamma LA-semiring S is a prime ideal. Definition 7. If I is a proper ideal of Γ -LA-semiring S and {0} 6= AΓB ⊆ I such that A ⊆ I or B ⊆ I for any ideals A,B of S, then I is called weakly prime ideal of gamma LA-semiring S. W. A. Khan et al. / Eur. J. Pure Appl. Math, 14 (3) (2021), 989-1001 995 Definition 8. If sn = 0 for s ∈ S and positive integer n (depending on s), then the element s in a Γ -LA-semiring S is nilpotent. The set of all nilpotent element of S is denoted by Nil S. Definition 9. If In = 0 for positive integer n (depending on I), then ideal I in a Γ -LA- semiring S is nilpotent. Theorem 7. Let A be a subtractive ideal in a Γ -LA-semiring S with 1 6= 0. Then the followings are equivalent. (i) A is a weakly prime ideal. (ii) If {0} 6= XΓY ⊆ A for right (left) ideals X,Y of S, then X ⊆ A or Y ⊆ A. (iii) If x, y ∈ S such that {0} 6= xΓSΓy ⊆ A, then x ∈ A or y ∈ A. Proof. (i)⇒ (ii) Let A be a weakly prime ideal of S and X,Y are two right (left) ideals of S such that {0} 6= XΓY ⊆ A. Let the ideals generated by X,Y are < X >,< Y >, respectively. Then {0} 6=< X > Γ < Y >⊆ A implies < X >⊆ A or < Y >⊆ A and X ⊆< X >⊆ A or Y ⊆⊆ A. Therefore, X ⊆ A or Y ⊆ A. (ii)⇒ (iii) Let {0} 6= xΓSΓy ⊆ A. Since S has an identity, therefore {0} 6= (xΓS)(yΓS) ⊆ A implies x ∈ xΓS ⊆ A or y ∈ yΓS ⊆ A. (iii) ⇒ (i) Suppose that XΓY ⊆ A, for ideals X and Y of S, where X 6⊆ A and Y 6⊆ A. Let x ∈ X\A, y ∈ Y \A. Also let x′ ∈ XnA, y′ ∈ Y nA be chosen arbitrary. Since x + x′, y + y′ 6∈ A, we must have {0} = (x + x′)ΓSΓ (y + y′). Now if we are let- ting x′ = 0 or y′ = 0 or x′ = 0 and y′ = 0 and considering all combinations we get 0 = xγy = x′γy = xγy′ = x′γy′ and hence XΓY = {0}. Proposition 1. Every ideal of a gamma LA-semiring S is weakly prime iff we have XΓY = X, XΓY = Y , or XΓY = 0, for any ideals X,Y in S. Proof. Assume X and Y are the weakly prime ideals of S. Suppose XΓY 6= S. Then XΓY is a weakly prime. If {0} 6= XΓY ⊆ XΓY , then we have X ⊆ XΓY or Y ⊆ XΓY (since XΓY is weakly prime ideal of S), that is, X = XΓY or Y = XΓY . If XΓY = S then we have X = Y = S, whence SAs = S. Conversely, let A be any proper ideal of S and let {0} 6= XΓY ⊆ A for ideals X and Y of S. Then, we have either X = XΓY ⊆ A or Y = XΓY ⊆ A. On the basis of above proposition we can easily prove the following results. Remark 3. If every ideal of Γ -LA-semiring S is a weakly prime, then we have either X2 = X or X2 = 0, for any ideal X of S. Lemma 4. Let P be a subtractive and weakly prime ideal but not a prime ideal of Γ -LA- semiring S. Let xγy = 0, for some x, y /∈ P , then we have xΓP = PΓy = {0}. Proof. Suppose xγp1 6= 0, for some p1 ∈ P and γ ∈ Γ . Then 0 6= xγ(y+p1) ∈ P . Since P is a weakly prime ideal of S, therefore y + p1 ∈ P or x ∈ P , that is, x ∈ P or y ∈ P , a contradiction. Therefore xΓP = {0}. Similarly, we can show that PΓy = {0}. W. A. Khan et al. / Eur. J. Pure Appl. Math, 14 (3) (2021), 989-1001 996 Theorem 8. Let P be a subtractive ideal of a Γ -LA-semiring S. If P is weakly prime but not a prime, then P 2 = {0}. Proof. Suppose p1γp2 6= 0, for some p1, p2 ∈ P and γ ∈ Γ and xγy = 0, for some x, y /∈ P , where P is not a prime ideal of S. Then by Lemma 4 we have (x+p1)γ(y+p2) = p1γp2 6= 0. Hence either (x+ p1) ∈ P or (y + p2) ∈ P , and thus either x ∈ P or y ∈ P , a contradiction. Hence P 2 = {0}. 3.1. IDEALS IN Γ-LA-SEMIRING In this section, we introduce the left and right ideals of Γ -LA-semiring and present some results on bi-ideal and quasi ideal in Γ -LA-semiring. Lemma 5. Let S be a gamma LA-semiring with identity. Then aγb = aβb, for all a, b ∈ S and γ, β ∈ Γ . Proof. Let S be a Γ -LA-semiring and e be the identity of S. Let x, y ∈ S and γ, β ∈ Γ . Then, we have xγy = xγ(eβy) = eγ(xβy) = xβy Lemma 6. Let S be a gamma LA semiring with identity and x ∈ S. If X is a left ideal of S then Xγx is a left ideal inS, whereγ ∈ Γ . Proof. If S is gamma LA-semiring having left identity and let x ∈ S. Now consider sγx+ rγx = (s+ r)γx ∈ Xγx. And SΓ (Xγx) ⊆ (SΓX)γx ⊆ Xγx for all r, s ∈ X and γ ∈ Γ . Hence Xγx is a left ideal inS. Corollary 2. Let S be a gamma LA-semiring with identity and x ∈ S. If X is a right ideal of S, then xγX is a right ideal in S, where γ ∈ Γ . Proof. It is similar to the proof of Lemma 6. Lemma 7. Let S be a gamma LA-semiring with identity and X,Y be the left ideals of S. then, for each left ideal Y of S, (X : Γ : Y ) is a left ideal in S, where (X : Γ : Y ) = {x ∈ S : xΓY ⊆ X}. W. A. Khan et al. / Eur. J. Pure Appl. Math, 14 (3) (2021), 989-1001 997 Proof. Suppose that S is a gamma LA-semiring with left identity. Let s ∈ S and let x, y ∈ (X : Γ : Y ). Then xΓY ⊆ X and yΓY ⊆ X so that (x+ y)ΓY = xΓY + yΓY ⊆ X +X = X. And (sγx)ΓY = sγ(xΓY ) ⊆ sγX ⊆ X for all γ ∈ Γ . Hence x + y ∈ (X : Γ : Y ) and SΓ (X : Γ : Y ) ⊆ (X : Γ : Y ). Thus (X : Γ : Y ) is a left ideal in S. Corollary 3. Let S be a gamma LA-semiring with identity and X be a left ideal of S. Then, (X : γ : r) is a left ideal in S, where (X : γ : r) = {x ∈ S : xγr ∈ X}. Proof. This follows from lemma 7 Remark 4. Let X,Y and Z be the left ideals of a gamma LA-semiring S. Then (X : Γ : Z) ⊆ (X : Γ : Y ), where Y ⊆ Z. Theorem 9. Let S be a Γ-LA-semiring with identity. Then, (X : Γ : Y ) is a quasi-ideal in S if X is quasi-ideal of S. Proof. Assume that X is a quasi-ideal of S, then By Lemma 7, we have (X : Γ : Y ) is a left ideal in S. Then, (SΓ (X : Γ : Y )) ∩ ((X : Γ : Y )ΓS) ⊆ (X : Γ : Y ) ∩ (X : Γ : Y ) ⊆ (X : Γ : Y ). Hence (X : Γ : Y ) is a quasi-ideal in S. Theorem 10. Let S be a gamma LA-semiring with identity. Then (X : Γ : Y ) is a left k-ideal in S, if X be a left k-ideal of S. Proof. Assume that X is a left k-ideal of S then By Lemma 7, (X : Γ : Y ) is a left ideal in S. Similarly, if x, x + t ∈ (X : Γ : Y ) then xΓY ⊆ X and (x + t)ΓY ⊆ X that is xΓY ⊆ X and xΓY + tΓY ⊆ X. Then, we get tΓY ⊆ X. Hence (X : Γ : Y ) is a left k-ideal in S. W. A. Khan et al. / Eur. J. Pure Appl. Math, 14 (3) (2021), 989-1001 998 3.2. ALMOST PRIME IDEALS IN Γ-LA-SEMIRING In this section, we initiate the term almost prime and weakly almost prime ideals in Γ -LA- semiring. Our starting point is the following definition. Definition 10. A left ideal P is called almost-prime if XΓY ⊆ P implies that X ⊆ P or Y ⊆ P , where X and Y are respectively left and right ideal of S. Remark 5. It is easy to see that every almost-prime left ideal is prime. Definition 11. A left ideal P is called weakly almost-prime if {0} 6= XΓY ⊆ P implies X ⊆ P or Y ⊆ P , where X and Y are respectively left and right ideal of S. Remark 6. It is easy to see that every almost-prime left ideal is weakly almost-prime. Lemma 8. Let P be the ideal of a Γ -LA-semiring S with identity. Then P is an almost- prime left ideal of S if xΓ (SΓy) ⊆ P implies x ∈ P or y ∈ P . Proof. Let P be an almost-prime left ideal of a Γ -LA-semiring S with identity. Now suppose that xΓ (SΓy) ⊆ P . Then by hypothesis, we have (SΓx)Γ (yΓS) ⊆ (SΓx)ΓSΓ (yΓS) = (xΓS)ΓSΓ (SΓy) = (SΓS)ΓxΓ (SΓy) ⊆ (SΓS)ΓP = (PΓS)ΓS ⊆ PΓS ⊆ P which implies (SΓx)Γ (yΓS) ⊆ P . Then, x = eγx ∈ SΓx ⊆ P or y = yγe ∈ yΓS ⊆ P . Hence x ∈ P ory ∈ P . Corollary 4. Let P be an almost-prime left ideal of a Γ -LA-semiring S with identity. Then P is a weakly almost-prime left ideal of S if {0} 6= xΓ (SΓy) ⊆ P , then x ∈ P or y ∈ P . Proof. This follows from Lemma 8 Theorem 11. Let S be a gamma LA-semiring with identity and x, y ∈ S and γ ∈ Γ . Then a left ideal P of S is almost-prime iff xγy ∈ P implies x ∈ P ory ∈ P . Proof. Let P be a left ideal of a Γ -LA-semiring with identity. Now suppose that xγy ∈ P , where x, y ∈ S and γ ∈ Γ . Then by hypothesis, we get; (SΓx)γ(yΓS) ⊆ SΓ ((xγy)ΓS) W. A. Khan et al. / Eur. J. Pure Appl. Math, 14 (3) (2021), 989-1001 999 ⊆ SΓ (PΓS) ⊆ SΓP ⊆ P. So by the definition of almost-prime, we have x ∈ P or y ∈ P . Conversely, assume that if xγy ∈ P implies x ∈ P or y ∈ P and X is left ideal of S. Let XΓY ⊆ P , where Y is right ideal of S such that Y ⊆ S−P . Then there exists y ∈ Y such that y /∈ P . Now we get xγy ∈ P . So by hypothesis, x ∈ P , for all x ∈ X =⇒ X ⊆ P . So P is almost-prime left ideal in S. Corollary 5. Let S be a gamma LA-semiring having identity and let x, y ∈ S, γ ∈ Γ . Then a left ideal P of S is weakly almost-prime iff 0 6= xγy ∈ P implies x ∈ P or y ∈ P . Proof. This follows from Theorem 11 Theorem 12. Let S be a gamma LA-semiring having left identity and X be an almost- prime left ideal of S. Then (X : Γ : Y ) is an almost-prime left ideal in S, where Y ⊆ S−X. Proof. Assume that X is a almost-prime left ideal of S. By Lemma 7, we have (X : Γ : Y ), a left ideal in S. Let xγy ∈ (X : Γ : Y ), where x, y ∈ S and γ ∈ Γ . Suppose that y /∈ (X : Γ : Y ). Since xγy ∈ (X : Γ : Y ), we have (xγy)ΓY ⊆ X. So by hypothesis (SΓx)γ(yΓY ) = SΓ ((xγy)ΓY ) ⊆ SΓX ⊆ X. Then, following the definition of almost-prime, we have x = eγx ∈ SΓx ⊆ X or yΓY ⊆ X implies that xΓS ⊆ XΓS ⊆ X. Hence (X : Γ : Y ) is an almost-prime left ideal in S. Corollary 6. Let S be a gamma LA-semiring having left identity and let X be an ideal of S. If X is a weakly almost-prime left ideal of S, then (X : Γ : Y ), is a weakly almost-prime left ideal in S, where Y ⊆ S −X. Proof. This follows from Theorem 12 Corollary 7. Let S be a gamma LA-semiring having left identity and let X be an ideal of S. If X is an almost-prime left ideal of S, then (X : γ : s) is an almost-prime left ideal in S, where s ∈ S −X and γ ∈ Γ . Proof. This follows from Theorem 12 Corollary 8. Let S be a gamma LA-semiring with left identity and let X be an ideal of S. If X is a weakly almost-prime left ideal of S, then (X : γ : s), is a weakly almost-prime left ideal in S, where s ∈ S −X and γ ∈ Γ . Proof. This follows from Corollary 5 REFERENCES 1000 4. Conclusion In this manuscript, firstly we have added some fresh examples along with new results in Γ -LA-rings. Next, we introduced the notion of Γ -LA-semirings and discussed different types of ideals in Γ -LA-semirings. We examined that almost all the results of LA-rings and LA-semirings are valid in case of Γ -LA-rings and Γ -LA-semirings. One could extend this work by shifting our results towards the theory of Γ -LA-nearrings, Γ -LA-hemirings etc. Acknowledgements Some results presented in this article are extracted from MS thesis of fourth author (Zahid Hussain) which was written under the supervision of the first author (Waheed Ahmad Khan) and was submitted to Higher Education of Pakistan. References [1] W. Barnes. On the gamma-rings of nobusawa. Pacific Journal of Mathematics, 18(3):411–422, 1966. [2] G. Booth and N. Groenewald. Equiprime gamma-near-rings. Quaestiones Mathemat- icae, 14:411–417, 1991. [3] J. R. Cho, J. Jezek Pusan, and T. Kepka. Praha, paramedial groupoids. Czechoslovak Mathematical Journal, 49:124, 1996. [4] D. M. Devi and G. S. Latha. La–semirings satisfying the identity ab= a+ b+ 1. International Journal of Innovative Science, Engineering & Technology, 2:378–389, 2015. [5] T. K. Dutta and S. K. Sardar. On prime ideals and prime radicals of a gamma- semirings. An. Stiint. Univ. Al. I. Cuza Iasi, Mat.(NS), 46:319–329, 2000. [6] T. K. Dutta and S. K. Sardar. Semiprime ideals and irreducible ideals of r-semirings. Novi Sad J. Math, 30(1):97–108, 2000. [7] R. D. Jagatap and Y. S. Pawar. Quasi-ideals and minimal quasi-ideals in gamma- semirings. Novi Sad J. Math, 39(2):79–87, 2009. [8] R. D. Jagatap and Y. S. Pawar. Quasi-ideals in regular gamma-semirings. Bulletin of Kerala Mathematics Association, 7(2):51–61, 2010. [9] Waheed Ahmad Khan, Abdelghani Taouti, Seema Karkain, Azar Salami, and Waqar Arif. Weakly prime and weakly primary ideals in gamma seminearrings. European Journal of Pure and Applied Mathematics, 12(2):544–552, 2019. REFERENCES 1001 [10] Q. Mushtaq and M. Khan. Ideals in la-semigroups. In Proc. of 4th International Pure Mathematics Conference, 2003. [11] Q. Mushtaq and S. M. Yusuf. On la-semi groups. the Alig. Bull.Math., 8:65–70, 1978. [12] Q. Mushtaq and S. M. Yusuf. On locally associative la-semigroups. J. Nat. Sci. Math, 19(1):57–62, 1979. [13] N. Nobusawa. On a generalization of the ring theory. Osaka Journal of Mathematics, 1(1):81–89, 1964. [14] M. M. K. Rao. Gamma-semirings-i. Southeast Asian Bull. Math, 19(1):49–54, 1995. [15] M. M. K. Rao. Ideals in ordered gamma-semirings. Discussiones Mathematicae- General Algebra and Applications, 38(1):47–68, 2018. [16] M. Sarwar. Conditions for LA-semigroups to resemble associative structures. PhD thesis, Quaid-i-Azam University Islamabad, 1993. [17] B. Satyanarayana. A note on gamma-near-rings. Indian J. Math, 41(3):427–433, 1999. [18] T Shah and I Rehman. On la-rings of finitely non-zero functions. Int. J. Contemp. Math. Sciences, 5(5):209–222, 2010. [19] P. Yiarayong. On left primary and weakly left primary ideals in gamma-la-rings. Gazi University Journal of Science, 29(1):143–148, 2016. [20] S. M. Yusuf. On left almost ring. In Proc. of 7th International Pure Math. Conference, Islamabad, 2006.