Prime and Maximal Ideals in Ternary Semigroups

This paper deals with prime and maximal ideals in ternary semigroup. In this paper we shall study the intersection of all prime ideals and intersection of all maximal ideals in a (non-commutative) ternary semigroup T. In particular we give a rather general necessary and sufficient condition in order that the set of all maximal ideals coincides with the set of all prime ideals. Prime and Maximal Ideals in Ternary Semigroups


Introduction
The relations between maximal and prime ideals in commutative rings are well known. If R is a ring, denote by N 0 the set of all nilpotent elements s∊R. We recall the following results.
• In any commutative ring the intersection of all the prime ideals is N 0 . • In any commutative ring with identity element any maximal ideal is prime. • If R is a commutative ring with identity element satisfying the descending chain condition every prime ideal of R is maximal.
Note explicitly that in a ring without an identity element maximal ideals need not be small prime and prime ideals need not be maximal [For this result, where some notions concerning semigroups are involved] 8 .
There are some reasons to have in mind the following analogy: Rings with an identity element ↔ ternary semigroups satisfying T 3 = T.
Rings without an identity element ↔ ternary semigroups in which T 3 ≠ T.

Definition 1
A non-empty ideal Q of a ternary semigroup T is said to be prime if ABC ⊂ Q implies that A ⊂ Q or B ⊂ Q or C ⊂ Q; A, B, C being ideals of T.
Remark: There is an analogous definition, an ideal Q is completely prime if abc ∊ Q implies that a ∊ Q or b ∊ Q or c ∊ Q; a, b, c ∊ T. An ideal which is completely prime. But the converse need not be true. These concepts coincide if T is commutative. In this paper we consider prime ideals in the sense of our definition prime ideals in the case of a ternary semigroup have been thoroughly studied in 7 .
Example 1: The ternary semigroup T itself is always a prime ideal of T. But T need not have prime ideals ≠ T.
Let e.g. be a ternary semigroup with zero in which x m = 0. Any ideal ≠ T is of the form I p = {a p ,…,a m-1 ,0}, 3 ≤ p ≤ m. Since we have I p ⊄ I p-1 and I p-1 I p-1 I p-1 ⊂ I p , I p is not a prime ideal of T.

Definition 2
An ideal M of T is known as a maximal if M is a proper ideal of T and is not properly contained in any proper ideal of T.
There are known some results concerning the existence of maximal ideals 1,2 . We shall not deal explicitly with these questions.
Example 2: The following example shows that a prime ideal need not be necessarily embeddable in a maximal ideal.
Denote by T 1 the multiplicative ternary semigroup of numbers x satisfying 0 ≤ x ≤ 1. Adjoin an element a and consider the set Then S is ternary semigroup and S 3 = S, S contains a unique maximal ideal, namely T 1 .
, a, T ® } is an ideal containing I, but clearly there does not exist a maximal ideal of S containing I.
In the following when speaking about maximal ideals we suppose, of course, that maximal ideal exists.

Theorem 1
If T is a ternary semigroup with T = T 3 , then every maximal ideal of T is prime ideal of T.
Proof: Let M be a maximal ideal of T. Denote T -M = P. We first prove P ⊂ P 3 , we have Since M P ∩ =φ , we have P ⊂ P 3 . Let now A, B, C be three ideals of T, none of them contained in M such that ABC ⊂ M. Since A ⊄ M and M is maximal, we have A ∪ M = T, hence P ⊂ A. By the same argument P ⊂ B and P ⊂ C. Hence P ABC P ABC 3 ⊂ ⊂ , . This contradicts ABC ⊂ M.
Remark: If T 3 ≠ T, then Theorem 1 does not hold. For, let a∊ T -T 3 . Then M =T -{a} is a maximal ideal of T and it is certainly not prime, since T 3 ⊂ M while T ⊄ M . But we can prove the following:

Theorem 2
If M is a maximal ideal of a ternary semigroup T such that T -M contains either more than one element or an idempotent then M is a prime ideal of T.
Proof: We shall use the following well known fact: If M is a maximal ideal of T, then the difference ternary semigroup T/M is simple and if T-M contains more than one element, then T/M cannot be nilpotent. Write again T = M ∪ P, M ∩ P = ∅ Let A, B, C be three ideals of T none of then contained in M such that ABC ⊂ M we again we have A ∪ M = B ∪ M = C ∪ M = T, hence P ⊂ A, P ⊂ B, P ⊂ C and P 3 ⊂ ABC; therefore P 3 ⊂ M . This would implies that T/M is nilpotent, which is impossible in both cases considered in the statement of our theorem.
Example 3: The following example serves to clarify the situation. Let S be the multiplicative ternary semigroup of numbers x [Note that any union of the type P A ∈  Q p ( A a subset of the set of all primes) is a prime ideal of T.] In contradiction to this we shall see in Theorem 3 that the intersection of all maximal ideals of any ternary semigroup is always non empty. In our example the maximal ideals are the sets M p = T-{p} and we have p We intend to clarify under which conditions prime ideals are maximal ideals. To this we first prove the following crucial theorem.

Theorem 3
Let {M α /α∊A} be the set of all different maximal ideals of a ternary semigroup T. Suppose that A ≥ 3 and denote P α = T -M α and M * = α  M α we then have If I is an ideal of T and I ∩ P α ≠ ∅ then P α ⊂ I. e) For α ≠ β we have P α P β ⊂ M * , so that M * is not empty.

Theorem 4
Let T be a ternary semigroup containing maximal ideals and let M * be the intersection of all maximal ideals of T.
Then every prime ideal of T containing M * and different from T is a maximal ideal of T. Proof: Let Q be a prime ideal of T containing M * and Q ≠ T. We use the notations of Theorem 3. By (d) we have