Original Article
Int. J. Chem. Sci, Volume: 15( 4)

Strongly Prime  -Semigroup

*Correspondence:
Jyothi V Department of Mathematics, KL University, Guntur, Andhra Pradesh, India Tel: 0863-2399999 E-mail: jyothi.mindspac[email protected]

Received: September 08, 2017; Accepted: October 23, 2017; Published: October 25, 2017

Citation: Jyothi V, Sarala Y, Madhusudhana Rao D, et al. Strongly Prime equation-Semigroup. Int J Chem Sci. 2017;15(4):206

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 equation-semigroup as a generalization of semigroup as follows. Various kinds of equation-semigroups have been widely studied by many authors [2-6].

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

Prime and semiprime ideals of equation-semigroups

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

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

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

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

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

Definition 1.4: For any ideal Q of a equation-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 aequation-semigroup S is said to be right primary if for any ideal U and V, U equation V implies U ⊆ m (Q) or V ⊆ Q.

Theorem 1.6: Let S be a equation-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, PC is an m-system and xPC . Then there exists an n-system N ⊆ PC 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 equation. Since P is primary, equation implies thatequation Since P is a semiprime ideal, P = m (P). Hence, U P or VP. 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 equation VP. Since P is a prime ideal, Un (P) or VP by theorem 2.6. Now our claim is that n (P)m (P). Let xn (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 xm (P) and therefore U n (P) or VP implies that Um (P) or VP. 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 equation-semigroups

Definition 2.1: Let S be a equation-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 equation-semigroup. If A is a subset of S, we defined a right α -annihilator of A to be a right ideal equation

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

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

Definition 2.4: A right β -insulator for aS* is a finite subset of S, Mβ(a) such that equation, for all

Definition 2.5: A left β -insulator for aS* is a finite subset of S, Mβ(a) such that equation for all equation

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

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

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

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

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

Theorem 2.11: Let S be a equation-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 equation Since S is right strongly prime, for every equation , there exists a right β-insulator Mβ(a) for a. Thenequationequation there existsequation whereequation Hence S is prime.

Conversely, suppose that S is prime. We have to prove that S is right strongly prime. Let equation and consider the collection of right α-annihilator ideals of the formequation 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 equation we can find an elementequation Since S is a prime equation-semigroup, it follows from 2.6 theorem, that there exists bSequation

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

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

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

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

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

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

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

Proof: Suppose that S is right strongly prime equation-semigroup. To prove L is right strongly prime equation-semigroup, it is enough to prove that every nonzero element in L has a right insulator. Letequation Then there existsequation such thatequation Since S is right strongly prime, for every equation , there exist an β-insulator for ∪iequation Hence for anyequationequation Now fixequation consider the collectionequation We shall prove thatequation is an insulator forequation It is enough to prove thatequation Thenequation We claim thatequationequation implies thatequation Thereforeequation

equation

equation

By equation Hence,equation Sinceequation 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 equation-semigroup.

Theorem 2.17: A equation-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 equation-semigroup. In order to prove that S is a strongly prime equation-semigroup, we shall prove that for every equation , every non-zero element in S has a right β -insulator. Letequation Since S is a left weakly semiprimeequation-semigroup, equation Since L is right strongly prime, there exists a right insulator equationequation Therefore, for anyequation Considerequation . We now claim thatequation is a β -insulator for x. It is enough to prove that for each. Let equation thenequation Thereforeequation Henceequation andequation so thatequationequation Since S is faithful L \ R bimodule, we have y=0. Since equation is arbitrary, for every equation , every non-zero element in S has a right β -insulator. Hence S is a right strongly prime equation-semigroup. Similarly, if R is a left strongly prime equation-semigroup then S is a left strongly prime equation-semigroup. Converse part follows from Theorem 2.16.

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

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

References