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Strongly Prime  -Semigroup

*Correspondence:

Jyothi V Department of Mathematics, KL University, Guntur, Andhra Pradesh, India Tel: 0863-2399999 E-mail: jyothi.mindspace@gmail.com

Abstract

The paper introduces the concepts of β-insulator and strongly prime-semigroups. Several characterizations of them are furnished.

Keywords

β Insulator; Strongly prime; Right and left α-annihilator

Introduction

The idea of general semigroups was developed by Anjaneyulu [1]. Saha defined -semigroup as a generalization of semigroup as follows. Various kinds of -semigroups have been widely studied by many authors [2-6].

this paper we introduce and study the structure of β-insulator and strongly prime -semigroups. In this paper many important results of strongly prime ideals in semigroups have been extended to strongly prime ideals in -semigroups.

Prime and semiprime ideals of -semigroups

Definition 1.1: A subset A of a -semigroup S is said to be an m–system if

Definition 1.2: A subset A of a -semigroup S is said to be an n – system if

Lemma 1.3: Let S be a -semigroup. An ideal A in S is semiprime if and only if A^C is an n – system.

Proof: Suppose that A is a semiprime ideal and let Then Since A is semiprime It implies that is an n – system.

Conversely, suppose is an n – system and let . We shall prove that Since is an n – system. <a> So that Hence Thus, A is a semiprime ideal.

Definition 1.4: For any ideal Q of a -semigroup S, we define n (Q) to the set of elements x such that every n – system containing x of S contains an element of Q.

Definition 1.5: An ideal Q in a-semigroup S is said to be right primary if for any ideal U and V, U V implies U ⊆ m (Q) or V ⊆ Q.

Theorem 1.6: Let S be a -semigroup for any right primary ideal P in S, the following are equivalent

(i) P is a prime ideal.

(ii) P=n (P).

(iii) P is a semiprime ideal.

Proof: (i) ⇒(ii) Let P be a prime ideal then P ⊆ n (P) is obvious. On the other hand, let x ∈ n (P) and suppose that x ∉P.

Since P is prime, P^C is an m-system and x∈ P^C . Then there exists an n-system N ⊆ P^C such that x ∈N. But N is disjoint from P , therefore x∉ n (P), which is a contradiction. Hence x ∈P, so that n (P) ⊆ P

(ii) ⇒ (iii) is obvious.

(iii) ⇒ (i) Suppose that P is a semiprime ideal. We have to prove that P is a prime ideal. Let U and V be any ideal in S with . Since P is primary, implies that Since P is a semiprime ideal, P = m (P). Hence, U ⊆P or V ⊆ P. Thus P is a prime ideal in S.

Theorem 1.7: For any ideal P in S, P is prime if and only if P is primary and semiprime.

Proof: Suppose that P is a prime ideal. We have to prove that P is primary. Let U and V be any ideal in S such that U V ⊆ P. Since P is a prime ideal, U ⊆ n (P) or V ⊆ P by theorem 2.6. Now our claim is that n (P) ⊆ m (P). Let x ∈ n (P) and S be any m – system containing x. Since is any m – system is an n – system, S is an n – system containing x. Since x ∈ n (P), S meets P . Hence x ∈ m (P) and therefore U ⊆ n (P) or V ⊆ P implies that U ⊆ m (P) or V ⊆ P. Hence P is a primary ideal. Since every prime ideal is a semiprime ideal, P is semiprime and hence primary ideal.

Conversely, suppose that P is primary and semiprime ideal. By theorem 1.6, P is a prime ideal.

Strongly prime -semigroups

Definition 2.1: Let S be a -semigroup. Let S is said to be semiprime if 0 is a semiprime ideal. S is said to be prime if (0) is a prime ideal.

Definition 2.2: Let S be a -semigroup. If A is a subset of S, we defined a right α -annihilator of A to be a right ideal

Definition 2.3: Let S be a -semigroup. If A is a subset of S, we defined a left α-annihilator of A to be a left ideal

We adopt the symbol S^* to denote the nonzero element of S.

Definition 2.4: A right β -insulator for a ∈ S^* is a finite subset of S, M_β(a) such that , for all

Definition 2.5: A left β -insulator for a∈S^* is a finite subset of S, M_β(a) such that for all

Definition 2.6: A -semigroup S is sad to be a right strongly prime if for every , each non zero element of S, has a right β -insulator, that is for every and a∈S^* , there is a finite subsest M_β(a) such that for , for all α∈implies b =0.

Definition 2.7: A -semigroup S is sad to be a left strongly prime if for every , each non zero element of S, has a left β -insulator, that is for every and a∈S^* , there is a finite subsest M_β(a) such that for

Definition 2.8: A -semigroup S is sad to be a left weakly semiprime -semigroup if

Definition 2.9: A -semigroup S is sad to be a right weakly semiprime -semigroup if

Definition 2.10: A -semigroup S is sad to be a weakly semiprime -semigroup if it is both left and right weakly semiprime.

Theorem 2.11: Let S be a -semigroup with D.C.C on annihilators then S is prime if S is strongly prime.

Proof: Suppose that S is right strongly prime. To prove S is prime, let Since S is right strongly prime, for every , there exists a right β-insulator M_β(a) for a. Then there exists where Hence S is prime.

Conversely, suppose that S is prime. We have to prove that S is right strongly prime. Let and consider the collection of right α-annihilator ideals of the form where I run over all finite subsets of S containing the identity. Since S satisfies the d.c.c. on right annihilators, choose a minimal element K. If we can find an element Since S is a prime -semigroup, it follows from 2.6 theorem, that there exists b∈S

Let be a finite subset of S containing the identity and b. Since, , a contradiction. This forces that Thus, s has a right β-insulator Since is arbitrary, every element of has a right β-insulator Similarly, every element of has a left β-insulator Hence S is a strongly prime - semigroup.

Definition 2.12: Let S be a -semigroup. A left ideal I of S is said to be essential if for all nonzero left ideals J of S.

Definition 2.13: The singular ideal of a -semigroup S is the ideal composed of elements whose right α-annihilator for each is an essential right ideal.

Theorem 2.14: If S is a strongly prime -semigroup having no zero devisor, then singular ideal is zero.

Proof: Let S is a strongly prime -semigroup and A be a singular ideal. Suppose that there exists an element such that Let M_β(a) be a right β -insulator for a. Since A is an ideal, Now implies that Then aβbα . Since A is singular, is essential for all We know that the intersection of finitely many essential right ideals is nonzero. Since M_β(a) is finite,, which is a contradiction to the β -insulator M_β(a) Consequently A=0.

Definition 2.15: Let S be a -semigroup. Let us define a relation if and only if for all Then ρ is an equivalence relation. Let [x, α ] denote the equivalence class containing Then L is a semigroup with respect to the multiplication defined by This semigroup L is called the left operator semigroup of the - semigroup.

Theorem 2.16: If S is a right strongly prime -semigroup, then the left operator -semigroup L(R) is right strongly prime -semigroup.

Proof: Suppose that S is right strongly prime -semigroup. To prove L is right strongly prime -semigroup, it is enough to prove that every nonzero element in L has a right insulator. Let Then there exists such that Since S is right strongly prime, for every , there exist an β-insulator for ∪_i Hence for any Now fix consider the collection We shall prove that is an insulator for It is enough to prove that Then We claim that implies that Therefore

By Hence, Since is arbitrary, every nonzero element in L has a right β -insulator. Similarly, if S is left strongly prime, then every non-zero element of R has a left β -insulator. Thus, L is right strongly prime, and R is a left strongly prime -semigroup.

Theorem 2.17: A -semigroup S is weakly semiprime then S is strongly prime and only if its left operator semigroup L is right strongly prime and its right operator semigroup R is left strongly prime.

Proof: Suppose that L is a right strongly prime -semigroup. In order to prove that S is a strongly prime -semigroup, we shall prove that for every , every non-zero element in S has a right β -insulator. Let Since S is a left weakly semiprime-semigroup, Since L is right strongly prime, there exists a right insulator Therefore, for any Consider . We now claim that is a β -insulator for x. It is enough to prove that for each. Let then Therefore Hence and so that Since S is faithful L \ R bimodule, we have y=0. Since is arbitrary, for every , every non-zero element in S has a right β -insulator. Hence S is a right strongly prime -semigroup. Similarly, if R is a left strongly prime -semigroup then S is a left strongly prime -semigroup. Converse part follows from Theorem 2.16.

Proposition 2.18: If S is strongly prime -semigroup, then S is weakly semiprime -semigroup.

Proof: Suppose that S is strongly prime -semigroup. We shall prove that S is a weakly semiprime -semigroup. Let It is enough to prove that Suppose that Since S is a strongly prime - semigroup, for every there exists a finite subset such that for implies that Hence x=0, a contradiction. Thus, S is a weakly semiprime -semigroup.

References

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