Ideal Theory in Commutative Ternary A-Semirings

International Mathematical Forum, Vol. 7, 2012, no. 42, 2085 - 2091

Ideal Theory in Commutative Ternary A-semirings

V. R. Daddi

Department of Mathematics D. Y. Patil College of Engineering and Technology Kolhapur, India vanita daddi@rediﬀmail.com

Y. S. Pawar

Department of Mathematics Shivaji University, Kolhapur, India pawar y [email protected]

Abstract

In this paper, we analyze some results on ideal theory of commutative ternary semirings with identity. We introduce and study the notion of ternary A-semiring.

Mathematics Subject Classiﬁcation: 16Y30, 16Y99

Keywords: Ternary semiring, ternary A-semiring, maximal ideal, prime ideal, semiprime ideal

1 Introduction

Algebraic structures play a prominent role in mathematics with wide ranging applications in many disciplines such as theoretical physics, computer sciences, control engineering, information sciences, coding theory, topological spaces and the like. This provides suﬃcient motivation to researchers to review various concepts and results. The theory of ternary algebraic system was introduced by D. H. Lehmer [1]. He investigated certain ternary algebraic systems called triplexes which turn out to be commutative ternary groups. The notion of ternary semigroups was introduced by Banach S. He showed by an example that a ternary semigroup does not necessarily reduce to an ordinary semigroups. In W.G. Lister [6] characterized additive subgroups of rings which are closed under the triple ring product and he called this algebraic system a 2086 V. R. Daddi and Y.S. Pawar ternary ring. T.K.Dutta and S. Kar ([3],[4],[5]) introduced and studied some properties of ternary semirings which is a generealization of ternary ring. Ternary semiring arises naturally as follows, consider the ring of integers Z which plays a vital role in the theory of ring. The subset Z+of all positive integers of Z is an additive semigroup which is closed under the ring product i.e. Z+is a semiring. Now,if we consider the subset Z−of all negative integers of Z, then we see that Z−is an additive semigroup which is closed under the triple ring product (however, Z− is not closed under the binary ring product), i.e. Z−forms a ternary semiring. Thus, we see that in the ring of integers Z, Z+ forms a semiring where as Z− forms a ternary semiring. Z+ forms a semiring where as Z− forms a ternary semiring in the fuzzy settings also. In this paper we will make an intensive study of the notions of prime ,com- pletely prime ideal in commutative ternary semirings. The notion of ternary A-semiring will be deﬁned which is generalization of A-semiring introduced by P.J.Allen and J.Neggers [2] and a characterization of ternary A-semiring will be presented. With the aid of these notions, further algebraic properties of the radical of an ideal in ternary A-semiring will be given.

2 Preliminary Notes

Deﬁnition 2.1. A non-empty set S together with a binary operation called addition and a ternary multiplication denoted by [], is said to be a ternary semiring if (S, +) is an additive commutative semigroup satisfying the following conditions :

1. [[abc] de]=[a [bcd] e]=[ab [cde]] ,

2. [(a + b) cd]=[acd]+[bcd] ,

3. [a (b + c) d]=[abd]+[acd] ,

4. [ab (c + d)] = [abc]+[abd] for all a, b, c, d, e ∈ S.

Now onwards unless stated otherwise S will denote a ternary semiring.

Deﬁnition 2.2. If there exists an element 0 ∈ S such that 0+x = x and [0xy]=[xy0] = [x0y]=0for all x, y ∈ S, then ”0” is called the zero element of S. In this case we say that S is a ternary semiring with zero.

Deﬁnition 2.3. An additive sub-semigroup T of S is called a ternary sub- semiring of S if [t1t2t3] ∈ T for all t1,t2,t3 ∈ T .

Deﬁnition 2.4. S is called commutative if [abc]=[bac]=[bca] for all a, b, c ∈ S. Ideal theory in commutative ternary A-semirings 2087

Deﬁnition 2.5. Let A, B, C be three subsets of S then by [ABC],wemean the set of all ﬁnite sums of the form [aibici] with ai ∈ A, bi ∈ B,ci ∈ C.

Deﬁnition 2.6. Let I be an additive subsemigroup of S. Then i) I is called a left ideal of S, if [s1s2i] ∈ I for all i ∈ I and for all s1,s2 ∈ S. ii) I is called a right ideal of S,if[is1s2] ∈ I for all i ∈ I and for all s1,s2 ∈ S. iii) I is called a lateral ideal of S, if [s1is2] ∈ I for all i ∈ I and for all s1,s2 ∈ S. iv) I is called an ideal of S, if I is a left, a right and a lateral ideal of S. An ideal I of S is called a proper ideal if I = S.

Deﬁnition 2.7. A proper ideal P of S is called a prime ideal of S,if [ABC] ⊆ P implies A ⊆ P or B ⊆ P or C ⊆ P for any three ideals A, B, C of S.

Deﬁnition 2.8. A proper ideal Q of S is called a semiprime ideal of S,if I3 ⊆ Q implies I ⊆ Q for any ideal I of S.

Deﬁnition 2.9. An ideal M in S is said to be maximal if i) M S and ii) If A is an ideal in S such that M A then A = S.

3 Ternary A-semirings

Deﬁnition 3.1. A non-empty subset T of S is closed with respect to ternary multiplication (t1,t2,t3 ∈ T implies [t1t2t3] ∈ T )ofS. An ideal P of S is said to be maximal with respect to T if, i) P ∩ T = ∅ and ii) If A is an ideal of S suct that P ⊂ A, then A ∩ T = ∅.

Theorem 3.2. Let T be a non-empty, ternary multiplicatively closed subset of S.IfA is an ideal of S such that A ∩ T = ∅, then there exists an ideal P of S such that A ⊂ P and P is maximal with respect to T .

Proof. Let denote the collection of all ideals in S which contain A and are disjoint from T . Since A ∈, it follows that = ∅. Deﬁne a relation ≤ on by B1 ≤ B2 ⇔ B1 ⊂ B2. is a partial ordered set under the relation ≤.If{Bi}i∈I is non-empty chain in , then ∪i∈I Bi ∈and Bi ≤∪i∈I Bi for every i ∈ I. Thus every non-empty chain in has an upper bound in , Zorn’s lemma implies that has a maximal element. Such a maximal element satisﬁes the conclusion of the theorem. 2088 V. R. Daddi and Y.S. Pawar

Theorem 3.3. Let T be a non-empty multiplicative closed subset of a commutative ternary semiring S and let P be an ideal in S.IfP is maximal with respect to T then P is prime. Proof. Let A, B and C be ideals in S such that P ⊂ A, P ⊂ B and P ⊂ C. Since P is maximal with respect to T , it follows that P ∩ T = ∅,A∩ T = ∅, B ∩ T = ∅ and C ∩ T = ∅. Let a ∈ A ∩ T, b ∈ B ∩ T, c ∈ C ∩ T. Since T is closed under multiplication it follows that [abc] ∈ [ABC] ∩ T , it follows that [ABC] P . Hence P is prime. Theorem 3.4. An ideal M in ternary semiring S with identity e is maximal if and only if it is maximal with respect to {e}. Proof. Suppose that M is maximal, it follows that M ∩{e} = ∅. Therefore there exits an ideal P in S such that P is maximal with respect to {e} and M ⊂ P . Clearly P ∩{e} = ∅ implies P S.Since M is maximal, it follows that M = P .ThusM is maximal with respect to { e }.

To prove converse, let us assume contrary that M is not maximal. Then there exists an ideal A in S such that M A S. Clearly A S implies A ∩{e} = ∅. Thus M is not maximal with respect to {e}, a contradiction. Hence M must be maximal. The following theorem is an immediate consequence of theorem 3.3 and theorem 3.4. Theorem 3.5. Let S be a commutative ternary semiring with identity e. If M is maximal ideal in S then M is prime. The following example shows that a maximal ideal in a commutative ternary semiring without an identity may not be prime. Example 3.6. Let S denote the ternary semiring of positive even integers with the usual addition and ternary multiplication. If M = {x ∈ S \ x>2} , then M is maximal ideal in S. Since 2 ∈/ M and 2.2.2=8∈ M. It follows that M is not prime.

Now we deﬁne ternary A-semiring. Deﬁnition 3.7. A ternary semiring S is said to be an ternary A-semiring if i) S is commutative. ii) Every proper ideal in S is contained in a prime ideal of S. Theorem 3.8. A commutative ternary semiring S is ternary A-semiring if and only if the complement of every proper ideal contains a non-empty multiplicatively closed set. Ideal theory in commutative ternary A-semirings 2089

Proof. If S is ternary A-semiring then every proper ideal in S is contained in a prime ideal of S. Hence it is clear that the complement of every proper ideal contains a non-empty multiplicatively closed set. Let B be a proper ideal in S and let T be a non-empty multiplicatively closed subset of S − B. Hence by theorem 3.2, there exists an ideal P in S such that B ⊂ P and P is maximal with respect to T and by theorem 3.3, P is prime.

Corollary 3.9. If S is a commutative ternary semiring with an identity e then S is ternary A-semiring.

Proof:If B is a proper ideal in S then {e}⊂S − B.

The following example will show there exists ternary A-semiring that do not have an identity.

Example 3.10. Let S be the ternary semiring of non-negative integers where a + b = max{a, b}, [abc] = min{a, b, c} Then S does not an identity and every proper ideal in S is prime.

Deﬁnition 3.11. Let B√ be a proper ideal in ternary A-semiring S. The radical of B is denoted by B and is deﬁned to be the intersection of all of the prime ideals in S that contains B.

The following theorem is an immediate consequence of deﬁnition 3.11. √ Theorem 3.12. If B is√ a proper ideal in ternary A-semiring S then B is an ideal in S and B ⊂ B.

Deﬁnition 3.13.√ A proper ideal B in ternary A-semiring S is said to be semi-prime if B = B.

Theorem 3.14. If B is a prime ideal in ternary A- semiring S, then B is semi-prime. √ B ⊂ B {P } Proof. Theorem 3.12 implies that . Let i i∈I be the√ collection of all B ∈{P } B ∩ P ⊂ prime ideals in S√ that contains B. Clearly, i i∈I and = i∈I i B . Thus, B = B.

The following example will show that there exists semi-prime ideals that are not prime. 2090 V. R. Daddi and Y.S. Pawar

Example 3.15. A ternary semiring Z+of non-negative integers is ternary A-semiring. Let (6) denote the ideal generated by 6 ∈ Z+.For1 ∈ Z+ it follows that (6) = {6n : n ∈ Z+}. Since 2 ∈/ (6), 3 ∈/ (6) and 2 · 2 · 3=12∈ (6). It is clear that (6)is not prime. The only prime ideals in Z+that contains (6) + + are (2), (3) and {0}∪{x ∈Z : x>1}. Since (2) ∩ (3) ∩{x ∈ Z : x>1} = (6). It follows that (6) = (6). Therefore (6) is semi-prime. √ Theorem 3.16. If B is a proper ideal in ternary A-semiring S, then B = {x ∈ S\∃ an odd integer n ∈ N such that xn ∈ B},where N denotes the set natural numbers. Proof. Let x ∈ S such that for some odd positive integer n it is valid that xn ∈ B. Let P be a prime ideal in S that contains B. Since P is prime and n x ∈ B ⊂ P , it follows that x ∈ √P . Since P is an arbitrary prime ideal in S containing B, it follows that x ∈ B. √ Conversely, assume there exists x ∈ B such that xn ∈/ B, for any odd integer n ∈ N. Choose one such x and let T = {xn\n is an odd positive integer }. Since T is a non-empty multiplicatively closed set in S such that T ∩ B = ∅. Hence theorem 3.2 implies that there exists an ideal P in S such that B ⊂ P and P is maximal with respect to T . Theorem 3.3 implies that√ P is prime. Since P ∩ T = ∅ and x ∈ T , it follows that x/∈ P .Thusx/∈ B,a contradiction. √ Corollary 3.17. If B is a proper ideal in ternary A-semiring then B is semi-prime. B S Proof. It√ is clear that√ the radical of√is a proper ideal in . Theorem 3.12 implies B ⊂ B . Let x ∈ B .Theorem√ 3.16 implies that there n exists an odd positive integer n such that x ∈ B. Also there exists√ an odd m xn m ∈ B xnm ∈ B x ∈ B positive√ integer√ such√ that ( ) .Thus and . Hence B ⊂ B.Therefore B is semi-prime.

Corollary 3.18. Let√A and√B be proper ideals in ternary A-semiring S. A ⊂ B A ⊂ B Then 1) If then√ .√ √ 2) If A ∩ B = ∅ then A ∩ B = A ∩ B. √ n Proof. (1)√ If x ∈ A, then there exists an n ∈ N such that x ∈ A ⊂ B. Hence x ∈ B. √ (2) Since A ∩ B = ∅, it is clear that A ∩ B is an ideal in S. Let x ∈ A ∩ B. n n Then there exists an n ∈ N such√ that x ∈ A √∩ B. Therefore x√∈ A√and n x ∈ B and it follows√ that√ x ∈ A and x ∈ B. Hence x ∈ A ∩ B. Consequently, x ∈ A ∩ B implies that there exists√ n, m ∈ N such that xn ∈ A and xm ∈ B. Clearly, xnm ∈ A ∩ B. Thus x ∈ A ∩ B. Ideal theory in commutative ternary A-semirings 2091

References

[1] D.H.Lehmer , A ternary analogue of abelian groups ; American Journal of Mathematics,5991932), 329-338

[2] P.J.Allen and J.Neggers, Ideal Theory in Commutative A-semirings, Kyungpook Math.J.46(2006),261-271.

[3] T.K.Dutta and S.Kar, On Prime Ideals And Prime Radical of Ternary Semirings,Bull.Cal.Math.Soc. 97(2005),445-454.

[4] T.K.Dutta and S.Kar, On Semiprime Ideals And Irreducible Ideals of Ternary Semirings, Bull.Cal.Math.Soc. 97(2005),467-476.

[5] T.K. Dutta and S. Kar , On regular ternary semirings, Advances in Alge- bra, Proceedings of the ICM Satellite Conference in Algebra and Related Topics,World Scientiﬁc, New Jersey, (2003), 343-355.

[6] W.G. Lister, Ternary Rings, Trans. Amer. Math. Soc., 154 (1971), 37-55.

Received: March, 2012