Original Article
, Volume: 15( 4)Strongly Prime ïÃÂÃâ¡ -Semigroup
- *Correspondence:
- Jyothi V Department of Mathematics, KL University, Guntur, Andhra Pradesh, India Tel: 0863-2399999 E-mail: jyothi.mindspace@gmail.com
Received: September 08, 2017; Accepted: October 23, 2017; Published: October 25, 2017
Citation: Jyothi V, Sarala Y, Madhusudhana Rao D, et al. Strongly Prime -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 -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 AC 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, PC is an m-system and x∈ PC . 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 . 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, Mβ(a) such that , for all
Definition 2.5: A left β -insulator for a∈S* is a finite subset of S, Mβ(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 Mβ(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 Mβ(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 Mβ(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 Mβ(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 Mβ(a) is finite,
, which is a contradiction to the β -insulator Mβ(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.
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