Prime Hyperideal in Multiplicative Ternary Hyperrings

International Journal of Algebra, Vol. 10, 2016, no. 5, 207-219 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ija.2016.6320 Prime Hyperideal in Multiplicative Ternary Hyperrings Md. Salim Department of Pure Mathematics University of Calcutta 35, Ballygunge Circular Road, Kolkata-700019, India T. K. Dutta Department of Pure Mathematics University of Calcutta 35, Ballygunge Circular Road, Kolkata-700019, India Copyright c 2016 Md. Salim and T. K. Dutta. This article is 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 In the year 1971, W. G. Lister [14] introduced the notion of ternary ring. Later in the year 2014, J. R. Castillo and Jocelyn S. Paradero- Vilela [2] introduced the notion of krasner ternary hyperring. In [6] we introduce the notion of multiplicative ternary hyperring. In this paper we study prime hyperideals in a multiplicative ternary hyperring. We obtain some Characterizations of prime hyperideal in a multiplicative ternary hyperring. Lastly we characterize regular strongly distributive multiplicative ternary hyperring with the help of prime hyperideal. Mathematics Subject Classification: 20N20 Keywords: prime hyperideal, strong m-system, weak m-system, maximal hyperideal, hyperintegral domain 1 Introduction Algebraic hyperstructure which is based on the notion of hyperoperation was introduced by F. Marty [1] in 1934 and studied extensively by many mathe-

208 Md. Salim and T. K. Dutta maticians.in [11] Corsini and Leoreanu-Fotea have collected numerous applications of algebraic hyperstructures from geometry, hypergraph, fuzzy sets and rough sets, cryptography and probabilities. The notion of multiplicative hyperring has been introduced by Rota [13] in which the addition is a binary operation and multiplication is a binary hyperoperation. Krasner s hyperring [3], introduced and studied by Krasner is a hypercompositional structure (S, +, ) where (S, +) is a cannonical hypergroup, (S, ) is a semigroup in which zero element is absorbing and the operation distributes from both sided over addition. In 1971, W.G.Lister[14] introduced the notion of ternary ring and study some important properties of it. According to Lister[14], a ternary ring is an algebraic system consisting of a non-empty set R together with a binary operation, called addition and a ternary multiplication, which forms a commutative group relative to addition, a ternary semigroup relative to multiplication and left, right, lateral distributive laws hold. In 2014, J. R. Castillo [2] introduced ternary hyperrings, called Krasner ternary hyperring. In Krasner ternary hyperring (R, +, ), + is a binary hyperoperation and is a ternary multiplication. In[6] we introduced the notion of multiplicative ternary hyperring. Our notion of multiplicative ternary hyperring differs from the notion of Krasner multiplicative ternary hyperring. In our multiplicative ternary hyperring (R, +, ), + is a binary operation and is a ternary hyperoperation, in which zero element is absorbing zero (i.e 0 R x y = x 0 R y = x y 0 R = {0 R } for all x, y R). Some earlier works on ternary ring and multiplicative ternary hyperring may be found in [4], [5], [7], [8] and [9]. In [12], Procesi and Rota introduced and studied the prime hyperideals in multiplicative hyperrings. The objective of this paper is to introduce and study prime hyperideal in a multiplicative ternary hyperring. We obtain some Characterizations of prime hyperideal of multiplicative ternary hyperring. We also introduce the notions of weak m-system and strong m-system and characterize of prime hyperideal with the help of these notions. Lastly as an application of prime hyperideal we characterize regular strongly distributive multiplicative ternary hyperring. 2 Preliminaries Definition 2.1. By a ternary hyperoperation on a nonempty set H, we shall mean a mapping : H H H P (H) when P (H) is the set of all nonempty subsets of H. For x, y, z H, the image of the element (x, y, z) H H H under the mapping will be denoted by x y z (which is called the ternary hyperproduct of x, y, z).

Prime hyperideal in multiplicative ternary hyperrings 209 Definition 2.2. A multiplicative ternary hyperring (R, +, ) is an additive commutative group (R, +) endowed with a ternary hyperoperation such that the following conditions hold : (i) (a b c) d e = a (b c d) e = a b (c d e); (ii) (a + b) c d a c d + b c d; a (b + c) d a b d + a c d; a b (c + d) a b c + a b d; (iii) ( a) b c = a ( b) c = a b ( c) = (a b c) for all a, b, c R; (iv) 0 R x y = x 0 R y = x y 0 R = {0 R } for all x, y R(absorbing property of 0 R ). for all a, b, c, d, e S, where if the inclusions in (ii) are replaced by equalities, then the multiplicative ternary hyperring is called a strongly distributive multiplicative ternary hyperring. Definition 2.3. A multiplicative ternary hyperring (R, +, ) is called commutative if a 1 a 2 a 3 = a σ(1) a σ(2) a σ(3), where σ is a permutation of {1, 2, 3} for all a 1, a 2, a 3 R. A multiplicative ternary hyperring (R, +, ) is commutative if and only if a b c = b a c = b c a for all a, b, c R. Definition 2.4. A nonempty subset ε = {(e i, f i )} n i=1 of R R where R is a multiplicative ternary hyperring is called a left (lateral, right) identity set of R if for any a R, a n i=1 e i f i a(resp. a n i=1 e i a f i, a n i=1 a e i f i ) A nonempty subset ε = {(e i, f i )} n i=1 of R R where R is a multiplicative ternary hyperring is called an identity set if it is a left and a lateral and a right identity set of R An element e of a multiplicative ternary hyperring (R, +, ) is called a unital element of R if a (e e a) (e a e) (a e e). Definition 2.5. Let (S, +, ) and (S, +, ) be two multiplicative ternary hyperrings. Then a mapping f : S S is called a homomorphism(a good homomorphism) if f(a + b) = f(a) + f(b) and f(a b c) f(a) f(b) f(c)(resp. f(a b c) = f(a) f(b) f(c)). Definition 2.6. Let (R, +, ) be multiplicative ternary hyperring. An additive subgroup I of R is called

210 Md. Salim and T. K. Dutta (i) a left hyperideal of R if r r x I, for all x I and for all r, r R; (ii) a right hyperideal of R if x r r I, for all x I and for all r, r R; (iii) a lateral hyperideal of R if r x r I, for all x I and for all r, r R; (iv) a two sided hyperideal of R if I is both a left and a right hyperideal of R; (v) a hyperideal of R if I is a left, a right, and a lateral hyperideal of R. Let R be a multiplicative ternary hyperring. If I, J and K are three nonempty subsets of R, then I J K = { a i b i c i : a i I, b i J, c i K}. Proposition 2.7. If (R, +, ) is a multiplicative ternary hyperring and a R, then a r = a R R + {na : n Z} is the right hyperideal of R generated by a. Similarly if (R, +, ) is a multiplicative ternary hyperring and a R, then a m = R a R + R R a R R + {na : n Z} is the lateral hyperideal of R generated by a, a l = R R a + {na : n Z} is the left hyperideal of R generated by a, and a = R R a + a R R + R a R + R R a R R + {na : n Z} is the hyperideal of R generated by a, where Z is the ring of integers. Proposition 2.8. If (R, +, ) is a multiplicative ternary hyperring with a unital element e or with an identity set, and a R then a r = a R R is the right hyperideal of R generated by a. Similarly a m = R R a R R is the lateral hyperideal of R generated by a. and a l = R R a is the left hyperideal of R generated by a, and is the hyperideal of R generated by a, a = R R a R R Remark 2.9. If (R, +, ) is a multiplicative ternary hyperring with a unital element e or with an identity set, then every lateral hyperideal is a left hyperideal as well as a right hyperideal. Hence in a multiplicative ternary hyperring with a unital element e or with an identity set the notions of hyperideal and lateral hyperideal coincide.

Prime hyperideal in multiplicative ternary hyperrings 211 3 Prime hyperideals of multiplicative ternary hyperrings Definition 3.1. A proper hyperideal P of a multiplicative ternary hyperring R is called a prime hyperideal of R if for any three hyperideals I, J, K of R; I J K P implies I P or J P or K P. Example 3.2. Consider the ring of integer Z, then Z is an additive commutative group. Let A = Z\3Z. Now we define a ternary hyperoperation on Z as follows a b c = {a x b y c : x, y A}. Then (Z, +, ) is a multiplicative ternary hyperring. We denote it by (Z A, +, ). Now 3Z is a hyperideal of (Z A, +, ). Let L, M, N be three hyperideals of (Z A, +, ) such that L M N 3Z. Let L 3Z and M 3Z then there exist elements l L and m M but l / 3Z and m / 3Z. Now for any n N, l m n 3Z. This implies that 3 l m n = l x m y n for x, y A. This implies that 3 n. So, n 3Z. Thus 3Z is a prime hyperideal of R. Theorem 3.3. Let (R, +, ) be a multiplicative ternary hyperring and P be a hyperideal of R. Then the following conditions are equivalent : (i) P is a prime hyperideal of R; (ii) a R b R c P, a R R b R R c P, a R R b R c R P and R a R b R R c P implies a P or b P or c P (iii) For any a, b, c R, a b c P a P or b P or c P, (iv) If U, V and W are right as well as lateral hyperideals of R such that U V W P then U P or V P or W P, (v) If U, V and W are left as well as lateral hyperideals of R such that U V W P then U P or V P or W P. Proof. (i) (ii) Suppose P is a prime hyperideal of R and a R b R c P, a R R b R R c P, a R R b R c R P, R a R b R R c P. Now (x 1 a y 1 ) (x 2 b y 2 ) (x 3 c y 3 ) = x 1 (a y 1 x 2 b y 2 x 3 c) y 3 x 1 (a R R b R R c) y 3 x 1 P y 3 P. Since P is a hyperideal of R, ( xi,yi x i a y i ) ( xj,yj x j b y j ) ( xk,yk x k c y k ) P. Hence ( { x i a y i }) ( { x j b y j }) ( { x k c y k }) P. i.e xi,yi xj,yj xk,yk (R a R) (R b R) (R c R) P. Then (R a R + R R a

212 Md. Salim and T. K. Dutta R R) (R b R + R R b R R) (R c R + R R c R R) R (a R R b R R c) R+R (a R b R c) R+R (R a R b R R c) R+R R (a R R b R c R)+R (a R R b R c R) R+(R a R b R R c) R R+ R R (a R b R c) R R+R R (a R R b R R c) R R R P R+R P R+R R P +R R P +R P R+P R R+R R P R R+R R P R R P. Since each of the above components are hyperideals of R, by primeness of P, it follows that (R a R+R R a R R) P or (R b R+R R b R R) P or (R c R + R R c R R) P Without loss of generality, suppose that (R a R + R R a R R) P. Then a a a = (R R a+a R R+R a R+R R a R R+{ma : m Z}) (R R a+a R R+R a R+R R a R R+{na : n Z}) (R R a+a R R+R a R+R R a R R+{pa : p Z}) (R a R+R R a R R) P, where m, n, p are integers. Since P is a prime hyperideal of R, a P and hence a P. Similarly if (R b R + R R b R R) P then b P and if (R c R + R R c R R) P then c P. (ii) (iii) Assume (ii) and a b c P for some a, b, c R Then a R b R c = a (R b R) c a b c P ; Similarly a R R b R R c P ; a R R b R c R P ; R a R b R R c P ; Then by using (ii), we find that a P or b P or c P. (iii) (iv) Suppose that U, V and W are three right as well as lateral hyperideals of R such that U V W P. Let U P and V P. Then there exists u U such that u / P and v V such that v / P. Let w W. Now u v w = (R R u+u R R+R u R+R R u R R+{mu : m Z}) (R R v + v R R + R v R + R R v R R + {nv : n Z}) (R R w + w R R + R w R + R R w R R + {pw : p Z}) (R R u+u) (R R v+v ) (R R w+w ) U V W +R R U V W P + R R P P + P P. By (iii) w w P,since u / P and v / P. This is true for all w W. So W P (iv) (i). Suppose (iv) holds and U, V, W be any three hyperideals of R such that U V W P. Then U, V and W are also right and lateral hyperideals of R. Hence by (iv) we have U P or V P or W P. Hence P is prime hyperideal of R. Dually we can prove that (iii) (v) (i). Corollary 3.4. Let (R, +, ) be a multiplicative ternary hyperring with a unital element e or with an identity set. Then a hyperideal P of R is prime if and only if a R R b R R c P implies a P or b P or c P ; Proof. Let P be a prime hyperideal of R. Suppose a R R b R R c P. Then a R b R c a R (e b e) R c a R R b R R c P. Similarly a R R b R c R P and R a R b R R c P. Now by above theorem a P or b P or c P. Similar is the proof when R contains identity set.

Prime hyperideal in multiplicative ternary hyperrings 213 Conversely, suppose that a R R b R R c P implies that a P or b P or c P where a, b, c R. Let U, V, W be three hyperideals of R such that U V W P. Let U P and V P. Then there exist elements u U and v V such that u / P and v / P. Now for any w W,u R R v R R w U V W P. By the given condition w P. Then W P. Hence P is a prime hyperideal of R. Corollary 3.5. A hyperideal P of a commutative multiplicative ternary hyperring R is prime if and only if a b c P implies that a P or b P or c P. Proof. Suppose P is a prime hyperideal of a commutative multiplicative ternary hyperring R and a b c P. Now a R (b R c) = a R (b c R) = a (R b c) R = (a b c) R R P R R P i.e a R b R c P, since R is commutative. Similarly a R R b R R c P, R a R b R R c P and a R R b R c R P. Since P is a prime hyperideal of R, now by (ii) of the theorem 3.3, we get a P or b P or c P. Converse is similar to the converse part of Corollary 3.4. Proposition 3.6. Let (R, +, ) be a multiplicative ternary hyperring and P be a prime hyperideal of R. Then for any a, b, c R, the following conditions are equivalent: (i) a b c P a P or b P or c P ; (ii) a b c P b a c P and a c b P. Proof. (i) (ii) It is obvious because P is a hyperideal of R. (ii) (i) Suppose that a b c P, where a, b, c R. Then (a b c) R R P R R P, since P is a hyperideal of R. Now for any y, x R, (a b c) y x P a (b c y) x P (a x b) y c P a x b y c P. This implies that a R b R c P. Again (a b c) R R R R (P R R) R R P R R P. Then for all s, t, x, y R, (a b c) s t x y P a (b c s t x) y P a (y b c s t) x P (a x y) (b c s) t P (a x y t b) c s P (a x y) t (b s c) P (a x y b s) c t P a x y b s t c P (by (ii)). This implies that a R R b R R c P. Similarly we can show that a R R b R c R P and R a R b R R c P. This implies that a P or b P or c P, by theorem 3.3(ii). In [10], N. H. McCoy introduced the notion of m-system as a generalization of multiplicative system and used to characterize prime ideal of rings. Here we generalize this notion for a multiplicative ternary hyperring and characterize prime hyperideal with the help of this notion.

214 Md. Salim and T. K. Dutta Definition 3.7. A non empty subset A of a multiplicative ternary hyperring (R, +, ) is called a weak m-system if for any a, b, c A, a R b R c A φ or a R R b R R c A φ or a R R b R c R A φ or R a R b R R c A φ. Definition 3.8. Let (R, +, ) be a multiplicative ternary hyperring. Then a nonempty subset A of R is called a strong m-system if there exist x, y, s, t R such that (a x b y c) A φ or (a x y b s t c) A φ or (a x y b s c t) A φ or (x a y b s t c) A φ. Remark 3.9. Every strong m-system is a weak m-system but the converse may not be true. Proposition 3.10. Let (R, +, ) be a multiplicative ternary hyperring and P be a hyperideal of R. Then the following conditions are equivalent : (i) P is a prime hyperideal of R. (ii) P c (complement of P in R) is a strong m-system. (iii) P c is a weak m-system. Proof. (i) (ii) Let a, b, c P c. Suppose that (a x b y c) P c = φ, for all x, y R. Then a x b y c P for all x, y R. This implies that a R b R c P. Next suppose that (a x y b s t c) P c = φ for all x, y, s, t R. Then a x y b s t c P for all x, y, s, t R. This implies that a R R b R R c P. Similarly if (a x y b s c t) P c = φ and (x a y b s t c) P c = φ for all x, y, s, t R then we can show that a R R b R c R P and R a R b R R c P respectively. Since P is prime, a P or b P or c P, a contradiction. Thus P c is a strong m-system. (ii) (iii) It is obvious. (iii) (i) Let a R b R c P, a R R b R R c P, a R R b R c R P and R a R b R R c P (A). If possible let a, b, c P c. Since P c is a weak m-system, (a R b R c) P c φ or (a R R b R R c) P c φ or (a R R b R c R) P c φ or (R a R b R R c) P c φ which contradicts (A). So a P or b P or c P. Hence P is a prime hyperideal of R. Theorem 3.11. If I is a hyperideal of a multiplicative ternary hyperring R and P is a prime hyperideal of R, then I P is a prime hyperideal of I, considering I as a multiplicative ternary hyperring. Proof. Obviously, I P is a hyperideal of I. Let a, b, c I\(I P ). Then a, b, c I and a, b, c / P. Hence a, b, c P c. Since P is a prime hyperideal of

Prime hyperideal in multiplicative ternary hyperrings 215 R, P c is a strong m-system. So for some x, y, s, t R, (a x b y c) P c φ or (a x y b s t c) P c φ or (a x y b s c t) P c φ or (x a y b s t c) P c φ. Suppose that (a x b y c) P c φ. Let r 1 (a x b y c) P c. Then r 1 P c. Also a, c P c. Since P c is a strong m-system there exist s, t R, (a s r 1 t c) P c φ. Since r 1 a x b y c, then a s r 1 t c a s a x b y c t c. Therefore (a s a x b y c t c) P c φ. Since s a x I and y c t I, (a I b I c) P c φ. Let r 2 (a I b I c) P c. Then r 2 / P r 2 / I P. Again r 2 a I b I c I. So r 2 I\(I P ). Hence (a I b I c) (I\(I P )) φ. Similarly if (a x y b s t c) P c φ or (a x y b s c t) P c φ or (x a y b s t c) P c φ, then we can show that (a I I b I I c) (I\(I P ) φ or (a I I b I c I) (I\(I P )) φ or (I a I b I I c) (I\(I P )) φ respectively. Thus I\(I P ) is a weak m-system in I. Hence by theorem 3.10 I P is a prime hyperideal of I. Definition 3.12. A hyperideal I of a multiplicative ternary hyperring (R, +, ) is called a maximal hyperideal of R if I R and for any hyperideal J I of R, I J R J = R. Proposition 3.13. Let I be a weak m-system of a multiplicative ternary hyperring (R, +, ). If P is a hyperideal which is maximal among all those hyperideals of R which are disjoint from I, then P is a prime hyperideal of R. Proof. If possible, let A, B, C be any three hyperideals of R such that A B C P, but A P, B P and C P. Now P A + P, P B + P and P C + P. Obviously (A + P ) I φ, (B + P ) I φ and (C + P ) I φ. So there exist i 1, i 2, i 3 I such that i 1 = a + p 1, i 2 = b + p 2 and i 3 = c + p 3 for some p 1, p 2, p 3 P, a A, b B and c C. Since I is a weak m-system and i 1, i 2, i 3 I, i 1 R i 2 R i 3 I φ or i 1 R R i 2 R R i 3 I φ or i 1 R R i 2 R i 3 R I φ or R i 1 R i 2 R R i 3 I φ. Suppose that i 1 R i 2 R i 3 I φ (A). Now for all r 1, r 2 R, i 1 r 1 i 2 r 2 i 3 = (a + p 1 ) r 1 (b + p 2 ) r 2 (c + p 3 ) a r 1 b r 2 c + p 1 r 1 b r 2 c + a r 1 p 2 r 2 c + p 1 r 1 p 2 r 2 c + a r 1 b r 2 p 3 + p 1 r 1 b r 2 p 3 + a r 1 p 2 r 2 p 3 + p 1 r 1 p 2 r 2 p 3 P. Thus i 1 R i 2 R i 3 P. So i 1 R i 2 R i 3 I P I = φ which contradicts (A). Similarly if we assume i 1 R R i 2 R R i 3 I φ or i 1 R R i 2 R i 3 R I φ or R i 1 R i 2 R R i 3 I φ we arrive at contradictions. So i 1 R i 2 R i 3 I = φ and i 1 R R i 2 R R i 3 I = φ and i 1 R R i 2 R i 3 R I = φ and R i 1 R i 2 R R i 3 I = φ which contradiction our assumption I is a weak m-system. So A P or B P or C P. Hence P is a prime hyperideal of R.

216 Md. Salim and T. K. Dutta Definition 3.14 (proposition 3.8, [7]). Let (R, +, ) be a multiplicative ternary hyperring and I be a hyperideal of R. Then the multiplicative ternary hyperring R/I = {a+i : a R} is called the quotient multiplicative ternary hyperring of (R, +, ) by I, where (a+i)+(b+i) = (a+b)+i and (a+i) (b+i) (c+i) = {p + I : p a b c} for any a, b, c R. Definition 3.15. Let (R, +, ) be a multiplicative ternary hyperring and I be a hyperideal of R. The mapping π : R R/I, defined by x x + I, is called the projection map of R by I. Theorem 3.16. The projection map π is a homomorphism of multiplicative ternary hyperrings. Proof. The proof is obvious. Definition 3.17. A multiplicative ternary hyperring (R, +, ) is called a multiplicative ternary hyperintegral domain if R is a commutative multiplicative ternary hyperring and a 1 a 2 a 3 = 0 implies that a 1 = 0 or a 2 = 0 or a 3 = 0, for all a 1, a 2, a 3 R. Theorem 3.18. If P R is a hyperideal in a commutative multiplicative ternary hyperring (R, +, ), then P is a prime hyperideal of R if and only if the quotient multiplicative ternary hyperring R/P is a multiplicative ternary hyperintegral domain. Proof. Obviously (R/P, +, ) is a commutative multiplicative ternary hyperring. Let P be a prime hyperideal and a 1 + P, a 2 + P and a 3 + P R/P for a 1, a 2, a 3 R be such that (a 1 + P ) (a 2 + P ) (a 3 + P ) = 0 + P = 0 R/P {y + P : y a 1 a 2 a 3 } = 0 + P = 0 R/P y + P = 0 + P y a 1 a 2 a 3 y P y a 1 a 2 a 3 a 1 a 2 a 3 P a 1 P or a 2 P or a 3 P, since P is a prime hyperideal of R a 1 + P = 0 + P or a 2 + P = 0 + P or a 3 + P = 0 + P Therefore R/P is a multiplicative ternary hyperintegral domain. Converse follows by reversing the above argument. 4 Characterization of regular multiplicative ternary hyperring by prime hyperideals Definition 4.1 ([9]). Let (R, +, ) be a multiplicative ternary hyperring. An element a R is said to be regular if there exists x R such that a a x a.

Prime hyperideal in multiplicative ternary hyperrings 217 The multiplicative ternary hyperring (R, +, ) is called regular if all of its elements are regular. Theorem 4.2. Let (R, +, ) be a commutative strongly distributive multiplicative ternary hyperring with a unital element e. Then R is regular if and only if every hyperideal of R, which is maximal with respect to the property of not having a given element in R is a prime hyperideal. Proof. Suppose that R is a regular commutative strongly distributive multiplicative ternary hyperring. Let P be a hyperideal of R, maximal with respect to the property of not containing a given element r R, then r / P (A). If P is not a prime hyperideal of R, then there exist three elements a, b, c R, a b c P, but a / P, b / P and c / P. Consider the hyperideal P + a. Then r P + a. So r p 1 + r i s i a for some p 1 P and r i, s i R. i Again Consider the hyperideal P + b. Then r p 2 + p i q i b for i some p 2 P and p i, q i R. Similarly, the element r is also an element of P + c. So r p 3 + u i v i c for some p 3 P and u i, v i R. Now i r r r (p 1 + i r i s i a) (p 2 + i p i q i b) (p 3 + i u i v i c) P, since R is commutative with a unital or an identity set. Since R is regular, there exists x R, r r x r (r x r) x r = (r r r) x x P, since R is commutative and r r r P. This is contradictions of (A). Hence P is a prime hyperideal of R. Conversely, assume that every hyperideal of R, maximal with respect the property of not having a given element, is a prime hyperideal. Let a R and I a = s R a a s. Using strongly distributive property we can show that I a is a hyperideal of R. Suppose that a / I a. Now Consider the collection M of hyperideals of R as follows: M = {J : J is a hyperideal of R, I a J and a / J}. Since I a M, the collection is nonempty. Then by the Zorn s lemma, M has a maximal element P. This hyperideal P is also maximal with respect to the property not having a. By hypothesis, P is a prime hyperideal. Now a a (a a a) I a P. So q a a P for all q a a a. Since

218 Md. Salim and T. K. Dutta a / P, we get q P for all q a a a. This implies that a a a P, since P is prime hyperideal of R, a P, a contradiction. Therefore a I a. This implies that a a a s a a s a for some s R. Thus the strongly distributive multiplicative ternary hyperring (R, +, ) is regular. References [1] F. Marty, Sur une generalization de la notion de group, 8 th Congress des Math. Scandenaves Stockholm, (1934), 45-49. [2] J. R. Castillo and Jocelyn S. Paradero-Vilela, Quotient and homomorphism in Krasner ternary hyperrings, International Journal of Mathematical Analysis, 8 (2014), no. 58, 2845-2859. http://dx.doi.org/10.12988/ijma.2014.410316 [3] M. Krasner, A Class of hyperrings and hyperfields, Internat. J. Math. Mathem. Sci., 6 (1983), 307-311. http://dx.doi.org/10.1155/s0161171283000265 [4] Md. Salim and T. K. Dutta, A Class of General Type of Regularities and Generalized Semiprime Ideals in Ternary Ring, accepted for publication in The Southeast Asian Bulletin of Mathematics. [5] Md. Salim Masud Molla, P. Mondal, T.K. Dutta, Subdirect Sum of Ternary Rings and Subdirectly Irreducible Ternary Rings, Journal of Progressive Research in Mathematics, 6 (2015), no. 2, 761-769. [6] Md. Salim, T. Chanda and T. K. Dutta, Regular equivalence and strongly regular equivalence on multiplicative ternary hyperrings, Journal of Hyperstructures, 4 (2015), no. 1, 20-36. [7] Md. Salim and T. K. Dutta, Correspondence between Hyperideals and Regular Congruence Relations of a Multiplicative Ternary Hyperring, (communicated). [8] Md. Salim and T. K. Dutta, On Right Strongly Prime Ternary Rings., (Accepted), East-West Journal of Mathematics. [9] Md. Salim and T. K. Dutta, Regular Multiplicative Ternary Hyperring, (communicated). [10] N. McCoy, Prime ideals in general rings, American Journal of Mathematics, 71 (1949), 823-833. http://dx.doi.org/10.2307/2372366

Prime hyperideal in multiplicative ternary hyperrings 219 [11] P. Corsini and V. Leoreanu, Applications of Hyperstructure Theory, vol. 5, Advances in Mathematics, Kluwer Academic Publishers, 2003. http://dx.doi.org/10.1007/978-1-4757-3714-1 [12] R. Procesi and R. Rota, On some classes of hyperstructures, combinatorics, Discrete Math., 1 (1997), 77-84. [13] R. Rota, Strongly distributive multiplicative hyperrings, Journal of Geometry, 39 (1990), no. 1, 130-138. http://dx.doi.org/10.1007/bf01222145 [14] W. G. Lister, Ternary rings, Trans. Amer. Math. Soc., 154 (1971), 37-55. http://dx.doi.org/10.1090/s0002-9947-1971-0272835-6 Received: March 30, 2016; Published: May 21, 2016