CHAPTER – 2

IDEALS IN - SEMIGROUPS

The theory of ideals in semigroups was studied by CLIFFORD and PRESTON

[12], [13]; PETRICH [39] and LJAPIN [31]. The ideal theory in commutative semigroups

was developed by BOURNE [7], HARBANS LAL [23], SATYANARAYANA [41], [42],

[43], [44], [45]; MANNEPALLI and NAGORE [38]. The ideal theory in general

semigroups was developed by ANJANEYULU [1], [2], [3], [4], [5], [40], GIRI and

WAZALWAR [22], HOEHNKE [24] and SCWARTZ [46]. SATYANARAYANA [43]

has developed some literature on prime ideals and prime radicals for commutative

semigroups. ANJANEYULU [1], [2], [3]; GIRI and WAZALWAR [22] studied about

prime radicals in general semigroups. The study of bi-ideals was made by AIYARED

IAMPAN [6], BRAJA ISLAM [9] in Γ-semigroups and by KAUSHIK and KHAN MOIN

[27] in Γ-semirings. IAMPAN[6] and BRAJA[9] introduced and characterized 0-minimal

and 0-maximal bi-ideals in Γ-semigroups. BRAJA ISLAM [8], [9]; JAGATAP and

PAWAR [25] studied about quasi ideals in Γ-semirings. CHINRAM [10] studied about

quasi-ideals and obtained some characterizations of regular Γ-semigroups. CHINRAM

and SIAMMAI [11] generalized the green’s relations in semigroups to Γ-semigroups and

to reductive Γ-semigroups. DHEENA and ELAVARASAN [14] made a study on prime

ideals, completely prime ideals, semiprime ideals and completely semiprime ideals in

partially ordered Γ-semigroups. MADHUSUDHANA RAO, ANJANEYULU and

GANGADHARA RAO [32], [33], [34] and [35] studied about the prime Γ-ideals,

completely prime Γ-ideals, semiprime Γ-ideals and completely semiprime Γ-ideals, prime

radicals in general Γ-semigroups. In this thesis we made a study about the Γ-ideals, prime

Γ-ideals, completely prime Γ-ideals, semiprime Γ-ideals and completely semiprime

Γ-ideals, prime radicals in Γ-semigroups.

This chapter is divided into 4 sections. In section 1, the terms; left Γ-ideal, right

Γ-ideal, Γ-ideal, proper Γ-ideal, trivial Γ-ideal, maximal left Γ-ideal, maximal right

Γ-ideal, maximal Γ-ideal, left Γ-ideal generated by a subset, right Γ-ideal generated by a

subset, Γ-ideal generated by a subset, principal left Γ-ideal, principal right Γ-ideal,

principal Γ-ideal of a Γ-semigroup are introduced. It is proved that (1) the nonempty

intersection of two left Γ-ideals of a Γ-semigroup S is a left Γ-ideal of S, (2) the nonempty

intersection of any family of left -ideals of a -semigroup S is a left -ideal of S, (3) the

Structure and ideal theory of duo Γ-semigroups Ideals in Γ-semigroups

55

union of two left -ideals of a -semigroup S is a left -ideal of S and (4) the union of any

family of left -ideals of a -semigroup S is a left -ideal of S. It is also proved that

(1) the nonempty intersection of two right Γ-ideals of a Γ-semigroup S is a right Γ-ideal of

S, (2) the nonempty intersection of any family of right -ideals of a -semigroup S is a

right -ideal of S, (3) the union of two right -ideals of a -semigroup S is a right -ideal

of S and (4) the union of any family of right -ideals of a -semigroup S is a right -ideal

of S. Further it is proved that (1) the nonempty intersection of two Γ-ideals of a

Γ-semigroup S is a Γ-ideal of S, (2) the nonempty intersection of any family of -ideals of

a -semigroup S is a -ideal of S, (3) the union of two -ideals of a -semigroup S is a

-ideal of S and (4) the union of any family of -ideals of a -semigroup S is a -ideal of

S. It is proved that if S is a Γ-semigroup and a ∈ S then (i) L(a) = a ∪ S a,

(ii) R(a) = a ∪ a S, (iii) J(a) = a ∪ aΓS ∪ SΓa ∪ SΓaΓS. If A and B are two Γ-ideals in a

Γ-semigroup S, then it is proved that Al (B) = {x ∈ S : < x >ΓB ⊆ A} and

Ar (B) = { x ∈ S : B Γ< x > ⊆ A} are Γ-ideals of S containing A.

In section 2, the terms; completely prime Γ-ideal, c-system, prime Γ-ideal,

m-system are introduced. It is proved that every -subsemigroup of a -semigroup is a

c-system. It is also proved that a -ideal P of a -semigroup S is completely prime if and

only if S\P is either a c-system or empty. It is proved that if P is a Γ- ideal of a

Γ-semigroup S, then the conditions (1) if A, B are Γ- ideals of S and AΓB⊆P then either

A⊆P or B⊆P, (2) if a, b ∈ S such that aΓS1Γb ⊆ P, then either a ∈ P or b ∈ P are

equivalent. It is also proved that every completely prime -ideal of a -semigroup S is a

prime -ideal of S. Further it is proved that a -ideal P of a -semigroup S is a prime

-ideal of S if and only if S\P is an m-system or empty. In a globally idempotent

Γ-semigroup, it is proved that every maximal Γ-ideal is a prime Γ-ideal. It is also proved

that a globally idempotent Γ-semigroup having a maximal Γ-ideal, contains semisimple

elements.

In section 3, the terms; completely semiprime Γ-ideal, d-system, semiprime

Γ-ideal, n-system are introduced. It is proved that (1) every completely prime Γ-ideal of a

Γ-semigroup is completely semiprime (2) every completely semiprime Γ-ideal of a

Γ-semigroup is semiprime, (3) every prime Γ-ideal of a Γ-semigroup is a semiprime

Γ-ideal. It is also proved that the nonempty intersection of a family of (1) completely

prime Γ-ideals of a Γ-semigroup is completely semiprime, (2) prime Γ-ideals of a

Structure and ideal theory of duo Γ-semigroups Ideals in Γ-semigroups

56

Γ-semigroup is semiprime. It is proved that a -ideal P of a -semigroup S is completely

semiprime iff S\P is a d-system of S or empty. It is also proved that an ideal Q of a

Γ-semigroup S is semiprime iff S\Q is either an n-system or empty. Further it is proved

that if N is an n-system in a Γ-semigroup S and a ∈ N, then there exists an m-system M of

S such that a ∈ M and M ⊆ N.

In section 4, the terms; prime Γ-radical rad A ( A ), completely prime Γ-radical

c.rad A of a Γ-ideal A in a Γ-semigroup are introduced. If A and B are two Γ-ideals of a

Γ-semigroup S, then it is proved that (i) A ⊆ B ⇒ √(A) ⊆ √(B) and c.rad A ⊆ c.rad B,

(ii) √(AΓB) =√(A∩B) =√(A) ∩ √(B) and c.rad (AΓB) = c.rad (A∩B) = c.rad (A)

∩ c.rad (B), (iii) √(√(A)) = √(A) and c.rad (c.rad A) = c.rad A. If A is a Γ-ideal in a

Γ-semigroup S then it is proved that rad A is a semiprime Γ-ideal and c.rad A is a

completely semiprime Γ-ideal of S. It is proved that a Γ- ideal Q of Γ-semigroup S is a

semiprime Γ-ideal of S iff √(Q) = Q. It is also proved that if A is a Γ-ideal of a

Γ-semigroup S, then √(A) is the smallest semiprime Γ-ideal of S containig A. It is proved

that if P is a prime -ideal of a -semigroup S, then √((P )n-1

P) = P for all n ∈ N. It is

proved that in a -semigroup S with identity there is a unique maximal -ideal M such that

√((M )n-1

M) = M for all n ∈ N. Further it is proved that if A is a -ideal of a -semigroup

S then √A = {x ∈ S: every m-system of S containing x meets A} i.e,

√A = {x ∈ S : M(x) ∩ A ≠ ∅}.

2.1. Γ-IDEALS

Ideals has greater importance in the theory of semigroups and Γ-semigroups. In

this section, the terms; left Γ-ideal, right Γ-ideal, Γ-ideal, proper Γ-ideal, trivial Γ-ideal,

maximal left Γ-ideal, maximal right Γ-ideal, maximal Γ-ideal, left Γ-ideal generated by a

subset, right Γ-ideal generated by a subset, Γ-ideal generated by a subset, principal left

Γ-ideal, principal right Γ-ideal, principal Γ-ideal of a Γ-semigroup are introduced. It is

proved that (1) the nonempty intersection of two left Γ-ideals of a Γ-semigroup S is a left

Γ-ideal of S, (2) the nonempty intersection of any family of left -ideals of a -semigroup

S is a left -ideal of S, (3) the union of two left -ideals of a -semigroup S is a left

-ideal of S and (4) the union of any family of left -ideals of a -semigroup S is a left

-ideal of S. It is also proved that (1) the nonempty intersection of two right Γ-ideals of a

Γ-semigroup S is a right Γ-ideal of S, (2) the nonempty intersection of any family of right

Structure and ideal theory of duo Γ-semigroups Ideals in Γ-semigroups

57

-ideals of a -semigroup S is a right -ideal of S, (3) the union of two right -ideals of a

-semigroup S is a right -ideal of S and (4) the union of any family of right -ideals of a

-semigroup S is a right -ideal of S. Further it is proved that (1) the nonempty

intersection of two Γ-ideals of a Γ-semigroup S is a Γ-ideal of S, (2) the nonempty

intersection of any family of -ideals of a -semigroup S is a -ideal of S, (3) the union of

two -ideals of a -semigroup S is a -ideal of S and (4) the union of any family of

-ideals of a -semigroup S is a -ideal of S. It is proved that if S is a Γ-semigroup and

a ∈ S then (i) L(a) = a ∪ S a, (ii) R(a) = a ∪ a S, (iii) J(a) = a ∪ aΓS ∪ SΓa ∪ SΓaΓS. If

A and B are two Γ-ideals in a Γ-semigroup S, then it is proved that

Al (B) = {x ∈ S : < x >ΓB ⊆ A} and A

r (B) = { x ∈ S : B Γ< x > ⊆ A} are Γ-ideals of S

containing A.

We now introduce the term of a left Γ-ideal in a Γ-semigroup.

DEFINITION 2.1.1 : A nonempty subset A of a Γ-semigroup S is said to be a left Γ-ideal

of S if , ,s S a A implies s a A .

The following note of a left Γ-ideal in a Γ-semigroup is due to JIROJKUL. CH,

SRIPAKORN. R and CHINRAM. R [26].

NOTE 2.1.2 : A nonempty subset A of a Γ-semigroup S is a left Γ- ideal of S iff SΓA⊆A.

We now characterize the left Γ- ideal of S.

THEOREM 2.1.3 : The nonempty intersection of any two left Γ-ideals of a

Γ-semigroup S is a left Γ-ideal of S.

Proof : Let A, B be two left Γ-ideals of S. Let a ∈ A∩B and s ∈ S, γ ∈ Γ.

a ∈ A∩B ⇒ a ∈ A and a ∈ B.

a ∈ A, s ∈ S, γ ∈ Γ, A is a left Γ-ideal of S ⇒ s a ∈ A.

a ∈ B, s ∈ S, γ ∈ Γ, B is a left Γ-ideal of S ⇒ s a ∈ B.

s a ∈ A, s a ∈ B ⇒ s a ∈ A∩B. Therefore A∩B is a left Γ-ideal of S.

THEOREM 2.1.4 : The nonempty intersection of any family of left -ideals of a

-semigroup S is a left -ideal of S.

Proof : Let { }A be a family of left Γ-ideals of S and let A A

.

Let a ∈ A, s ∈ S, γ ∈ Γ.

Structure and ideal theory of duo Γ-semigroups Ideals in Γ-semigroups

58

a ∈ A ⇒ a ∈ A

⇒ a A for each α ∈ Δ.

a A , s ∈ S, γ ∈ Γ, A is a left Γ-ideal of S ⇒ s a ∈ A .

s a ∈ A for all α ∈ Δ ⇒ s a ∈ A

⇒ s a ∈ A. Therefore A is a left Γ-ideal of S.

THEOREM 2.1.5 : The union of any two left -ideals of a -semigroup S is a left

-ideal of S.

Proof : Let A1, A2 be two left Γ-ideals of a Γ-semigroup S. Let A = A1∪A2.

Clearly A is a nonempty subset of S. Let a ∈ A, s ∈ S and γ ∈ Γ.

a ∈ A ⇒ a ∈ A1∪A2 ⇒ a ∈ A1 or a ∈ A2.

If a ∈ A1 then a ∈ A1, s ∈ S, γ ∈ Γ, A1 is a left Γ-ideal of S ⇒ s a ∈ A1 ⊆ A1∪A2 = A

⇒ s a ∈ A.

If a ∈ A2 then a ∈ A2, s ∈ S, γ ∈ Γ, A2 is a left Γ-ideal of S ⇒ s a ∈ A2 ⊆ A1∪A2 = A

⇒ s a ∈ A.

Therefore a ∈ A, s ∈ S, γ ∈ Γ ⇒ s a ∈ A and hence A is a left Γ-ideal of S.

THEOREM 2.1..6 : The union of any family of left -ideals of a -semigroup S is a

left -ideal of S.

Proof : Let { }A be a family of left Γ-ideals of a Γ-semigroup S and let A A

.

Clearly A is a nonempty subset of S. Let a ∈ A, s ∈ S, ∈ Γ.

a ∈ A ⇒ a ∈ A

⇒ a ∈ Aα for some ∈ Δ.

a ∈ Aα, s ∈ S, ∈ Γ, Aα is a left Γ-ideal of S ⇒ s a ∈ Aα ⊆ A

= A ⇒ s a ∈ A.

Therefore A is a left Γ-ideal of S.

We now introduce the notion of a right Γ-ideal in a Γ-semigroup.

DEFINITION 2.1.7 : A nonempty subset A of a Γ-semigroup S is said to be a right

Γ-ideal of S if , ,s S a A implies a s A .

The following note of a right Γ-ideal in a Γ-semigroup is due to JIROJKUL. CH,

SRIPAKORN. R and CHINRAM. R [26].

Structure and ideal theory of duo Γ-semigroups Ideals in Γ-semigroups

59

NOTE 2.1.8 : A nonempty subset A of a Γ-semigroup S is a right Γ- ideal of S iff

AΓS⊆A.

We now characterize the right Γ- ideal of S.

THEOREM 2.1.9 : The nonempty intersection of any two right Γ-ideals of a

Γ-semigroup S is a right Γ-ideal of S.

Proof : Let A, B be two right Γ-ideals of S. Let a ∈ A∩B, s ∈ S and γ ∈ Γ.

a ∈ A∩B ⇒ a ∈ A and a ∈ B.

a ∈ A, s ∈ S, γ ∈ Γ, A is a right Γ-ideal of S ⇒ a s ∈ A.

a ∈ B, s ∈ S, γ ∈ Γ, B is a right Γ-ideal of S ⇒ a s ∈ B.

a s ∈ A, a s ∈ B ⇒ a s ∈ A∩B and hence A∩B is a right Γ-ideal of S.

THEOREM 2.1.10 : The nonempty intersection of any family of right -ideals of a

-semigroup S is a right -ideal of S.

Proof : Let { }A be a family of right Γ-ideals of S and let A A

.

Let a ∈ A, s ∈ S, γ ∈ Γ.

a ∈ A ⇒ a ∈ A

⇒ a A for each α ∈ Δ.

a A , s ∈ S, γ ∈ Γ, A is a right Γ-ideal of S ⇒ a s ∈ A .

a s ∈ A for all α ∈ Δ ⇒ a s ∈ A

⇒ a s ∈ A. Therefore A is a right Γ-ideal of S.

THEOREM 2.1.11 : The union of any two right -ideals of a -semigroup S is a right

-ideal of S.

Proof : Let A1, A2 be two right Γ-ideals of a Γ-semigroup S. Let A = A1∪A2.

Clearly A is a nonempty subset of S. Let a ∈ A, s ∈ S, γ ∈ Γ.

a ∈ A ⇒ a ∈ A1∪A2 ⇒ a ∈ A1 or a ∈ A2.

If a ∈ A1 then a ∈ A1, s ∈ S, γ ∈ Γ, A1 is a right Γ-ideal of S

⇒ a s ∈ A1 ⊆ A1∪A2 = A ⇒ a s ∈ A.

If a ∈ A2 then a ∈ A2, s ∈ S, γ ∈ Γ, A2 is a right Γ-ideal of S

⇒ a s ∈ A2 ⊆ A1∪A2 = A ⇒ a s ∈ A.

∴ a ∈ A, s ∈ S, γ ∈ Γ then a s ∈ A. Therefore A is a right Γ-ideal of S.

Structure and ideal theory of duo Γ-semigroups Ideals in Γ-semigroups

60

THEOREM 2.1.12 : The union of any family of right -ideals of a -semigroup S is a

right -ideal of S.

Proof : Let { }A be a family of right Γ-ideals of S and let A A

.

Clearly A is a nonempty subset of S. Let a ∈ A, s ∈ S, ∈ Γ.

a ∈ A ⇒ a ∈ A

⇒ a ∈ Aα for some ∈ Δ.

a ∈ Aα, s ∈ S, ∈ Γ, Aα is a right Γ-ideal of S ⇒ a s ∈ Aα ⊆ A

= A ⇒ a s ∈ A.

Therefore A is a right Γ-ideal of S.

We now introduce the notion of a Γ-ideal of a Γ-semigroup.

DEFINITION 2.1.13 : A nonempty subset A of a Γ-semigroup S is said to be a two sided

Γ- ideal or simply a Γ- ideal of S if s ∈ S, a ∈ A, ∈ Γ imply s a ∈ A, a s ∈ A.

The following note of a Γ-ideal in a Γ-semigroup is due to JIROJKUL. CH,

SRIPAKORN. R and CHINRAM. R [26].

NOTE 2.1.14 : A nonempty subset A of a Γ-semigroup S is a two sided Γ-ideal iff it is

both a left Γ-ideal and a right Γ- ideal of S.

EXAMPLE 2.1.15 : Let N be the set of natural numbers and Γ = 2N. Then N is a

Γ-semigroup and A = 3N is a Γ- ideal of the Γ-semigroup N.

THEOREM 2.1.16 : The nonempty intersection of any two Γ-ideals of a Γ-semigroup

S is a Γ-ideal of S.

Proof : Let A, B be two Γ-ideals of S. Let a ∈ A∩B and s ∈ S, γ ∈ Γ.

a ∈ A∩B ⇒ a ∈ A and a ∈ B.

a ∈ A, s ∈ S, γ ∈ Γ, A is a Γ-ideal of S ⇒ s a, a s ∈ A.

a ∈ B, s ∈ S, γ ∈ Γ, B is a Γ-ideal of S ⇒ s a, a s ∈ B.

∴ s a, a s ∈ A, s a, a s ∈ B ⇒ s a, a s ∈ A∩B.

Therefore A∩B is a Γ-ideal of S.

THEOREM 2.1.17 : The nonempty intersection of any family of -ideals of a

-semigroup S is a -ideal of S.

Structure and ideal theory of duo Γ-semigroups Ideals in Γ-semigroups

61

Proof : Let { }A be a family of Γ-ideals of S and let A A

.

Let a ∈ A, s ∈ S, γ ∈ Γ.

a ∈ A ⇒ a ∈ A

⇒ a A for each α ∈ Δ.

a A , s ∈ S, γ ∈ Γ, A is a Γ-ideal of S ⇒ s a, a s ∈ A .

s a, a s ∈ A for all α ∈ Δ ⇒ s a, a s ∈ A

⇒ s a, a s ∈ A.

Therefore A is a Γ-ideal of S.

THEOREM 2.1.18 : The union of any two -ideals of a -semigroup S is a -ideal

of S.

Proof : Let A1, A2 be two Γ-ideals of a Γ-semigroup S. Let A = A1∪A2.

Clearly A is a nonempty subset of S. Let a ∈ A, s ∈ S, γ ∈ Γ.

a ∈ A ⇒ a ∈ A1∪A2 ⇒ a ∈ A1 or a ∈ A2.

If a ∈ A1 then a ∈ A1, s ∈ S, γ ∈ Γ, A1 is a Γ-ideal of S ⇒ s a, a s ∈ A1 ⊆ A1∪A2 = A

⇒ s a, a s ∈ A.

If a ∈ A2 then a ∈ A2, s ∈ S, γ ∈ Γ, A2 is a Γ-ideal of S ⇒ s a, a s ∈ A2 ⊆ A1∪A2 = A

⇒ s a, a s ∈ A.

Thus a ∈ A, s ∈ S, γ ∈ Γ ⇒ s a, a s ∈ A. Therefore A is a Γ-ideal of S.

THEOREM 2.1.19 : The union of any family of -ideals of a -semigroup S is a

-ideal of S.

Proof : Let { }A be a family of Γ-ideals of S and let A A

.

Clearly A is a nonempty subset of S. Let a ∈ A, s ∈ S, ∈ Γ.

a ∈ A ⇒ a ∈ A

⇒ a ∈ Aα for some ∈ Δ.

Now a ∈ Aα, s ∈ S, ∈ Γ, Aα is a Γ-ideal of S ⇒ s a, a s ∈ Aα ⊆ A

= A

Thus a ∈ A, s ∈ S, ∈ Γ ⇒ s a, a s ∈ A. Therefore A is a Γ-ideal of S.

We now introduce a proper Γ-ideal, trivial Γ-ideal, maximal left Γ-ideal, maximal

right Γ-ideal, maximal Γ-ideal and globally idempotent Γ-ideal of a Γ-semigroup.

DEFINITION 2.1.20 : A Γ-ideal A of a Γ-semigroup S is said to be a proper Γ-ideal of S

if A is different from S.

Structure and ideal theory of duo Γ-semigroups Ideals in Γ-semigroups

62

DEFINITION 2.1.21 : A Γ-ideal A of a Γ-semigroup S is said to be a trivial Γ-ideal

provided S\A is singleton.

DEFINITION 2.1.22 : A Γ-ideal A of a Γ-semigroup S is said to be a maximal left

-ideal provided A is a proper left Γ-ideal of S and is not properly contained in any

proper left Γ-ideal of S.

DEFINITION 2.1.23 : A Γ-ideal A of a Γ-semigroup S is said to be a maximal right

-ideal provided A is a proper right Γ-ideal of S and is not properly contained in any

proper right Γ-ideal of S.

DEFINITION 2.1.24 : A Γ-ideal A of a Γ-semigroup S is said to be a maximal -ideal

provided A is a proper Γ-ideal of S and is not properly contained in any proper

Γ-ideal of S.

DEFINITION 2.1.25 : A Γ-ideal A of a Γ-semigroup S is said to be globally idempotent

if AΓA = A.

THEOREM 2.1.26 : If A is a -ideal of a -semigroup S with unity 1 and 1 ∈ A then

A = S.

Proof : Clearly A ⊆ S. Let s ∈ S.

1 ∈ A, s ∈ S, A is a Γ-ideal of S ⇒ 1Γs ⊆ A ⇒ s ∈ A. Thus S ⊆ A.

A ⊆ S, S ⊆ A ⇒ S = A.

THEOREM 2.1.27 : If S is a -semigroup with unity 1 then the union of all proper

-ideals of S is the unique maximal -ideal of S.

Proof : Let M be the union of all proper Γ-ideals of S. Since 1 is not an element of any

proper Γ-ideal of S, 1 ∉ M. Therefore M is a proper subset of S. By theorem 2.1.19, M is

a Γ-ideal of S. Thus M is a proper Γ-ideal of S. Since M contains all proper Γ-ideals of S,

M is a maximal Γ-ideal of S. If M1 is any maximal Γ-ideal of S, then M1 ⊆ M ⊂ S and

hence M1 = M. Therefore M is the unique maximal Γ-ideal of S.

We now introducing left Γ-ideal generated by a subset, right Γ-ideal generated by a

subset, Γ-ideal generated by a subset of a Γ-semigroup.

DEFINITION 2.1.28 : Let S be a Γ-semigroup and A be a nonempty subset of S. The

smallest left Γ-ideal of S containing A is called left Γ-ideal of S generated by A.

Structure and ideal theory of duo Γ-semigroups Ideals in Γ-semigroups

63

THEOREM 2.1.29 : The left Γ-ideal of a Γ-semigroup S generated by a nonempty

subset A is the intersection of all left Γ-ideals of S containing A.

Proof : Let Δ be the set of all left Γ-ideals of S containing A.

Since S itself is a left Γ-ideal of S containing A, S ∈ Δ. So Δ ≠ ∅.

Let T

T T

. Since A ⊆ T for all T ∈ Δ, A T .

By theorem 2.1.5, T* is a left Γ-ideal of S.

Let K is a left Γ-ideal of S containing A.

Clearly A ⊆ K and K is a left Γ-ideal of S.

Therefore K ∈ Δ ⇒T* ⊆ K.

Therefore T* is the left Γ-ideal of S generated by A.

DEFINITION 2.1.30 : Let S be a Γ-semigroup and A be a nonempty subset of S. The

smallest right Γ-ideal of S containing A is called right Γ-ideal of S generated by A.

THEOREM 2.1.31 : The right Γ-ideal of a Γ-semigroup S generated by a nonempty

subset A is the intersection of all right Γ-ideals of S containing A.

Proof : Let Δ be the set of all right Γ-ideals of S containing A.

Since S itself is a right Γ-ideal of S containing A, S ∈ Δ. So Δ ≠ ∅.

Let T

T T

. Since A ⊆ T for all T ∈ Δ, A T .

By theorem 2.1.10, T* is a right Γ-ideal of S.

Let K is a right Γ-ideal of S containing A.

Clearly A ⊆ K and K is a right Γ-ideal of S.

Therefore K ∈ Δ ⇒T* ⊆ K.

Therefore T* is the right Γ-ideal of S generated by A.

DEFINITION 2.1.32 : Let S be a Γ-semigroup and A be a nonempty subset of S. The

smallest Γ-ideal of S containing A is called Γ-ideal of S generated by A.

THEOREM 2.1.33 : The Γ-ideal of a Γ-semigroup S generated by a nonempty subset

A is the intersection of all Γ-ideals of S containing A.

Proof : Let Δ be the set of all Γ-ideals of S containing A.

Since S itself is a Γ-ideal of S containing A, S ∈ Δ. So Δ ≠ ∅.

Structure and ideal theory of duo Γ-semigroups Ideals in Γ-semigroups

64

Let T

T T

. Since A ⊆ T for all T ∈ Δ, A T .

By theorem 2.1.17, T* is a Γ-ideal of S.

Let K is a Γ-ideal of S containing A.

Clearly A ⊆ K and K is a Γ-ideal of S.

Therefore K ∈ Δ ⇒T* ⊆ K.

Therefore T* is the Γ-ideal of S generated by A.

We now introduce a principal left Γ-ideal of a Γ-semigroup.

DEFINITION 2.1.34 : A left Γ-ideal A of a Γ-semigroup S is said to be the principal left

Γ-ideal generated by a if A is a left Γ-ideal generated by {a} for some a ∈ S. It is denoted

by L(a).

Now we characterize principal left Γ-ideal of a Γ-semigroup.

THEOREM 2.1.35 : If S is a -semigroup and a ∈ S then L(a) = a ∪ S a.

Proof : Let s ∈ S, r ∈ a ∪ SΓa and γ ∈ Γ.

r ∈ a ∪ SΓa ⇒ r = a or r = t a for some t ∈ S, α ∈ Γ.

If r = a then s r = s a ∈ SΓa ⊆ a ∪ SΓa.

If r = t a then s r = s (t a) = (s t) a ∈ SΓa ⊆ a ∪ SΓa.

Therefore s a ∈ a ∪ SΓa and hence a ∪ SΓa is a left Γ-ideal of S.

Let L be a left Γ-ideal of S containing a.

Let r ∈ a ∪ SΓa. Then r = a or r = t a for some t ∈ S, α ∈ Γ.

If r = a then r = a ∈ L. If r = t a then r = t a ∈ L.

Therefore a∪SΓa ⊆ L and hence a∪SΓa is the smallest left Γ-ideal containing a.

Therefore L(a) = a∪SΓa.

NOTE 2.1.36 : If S is a Γ-semigroup and a ∈ S then L (a) = S1Γa.

We now introduce principal right Γ-ideal of a Γ-semigroup.

DEFINITION 2.1.37 : A right Γ-ideal A of a Γ-semigroup S is said to be the principal

right Γ-ideal generated by a if A is a right Γ-ideal generated by {a} for some a ∈ S. It is

denoted R(a).

Now we characterize principal right Γ-ideal of a Γ-semigroup.

THEOREM 2.1.38 : If S is a -semigroup and a ∈ S then R(a) = a ∪ a S.

Structure and ideal theory of duo Γ-semigroups Ideals in Γ-semigroups

65

Proof : Let s ∈ S, r ∈ a ∪ aΓS.

Now r ∈ a ∪ aΓS ⇒ r = a or r = a t for some t ∈ S, α ∈ Γ.

If r = a then r s = a s ∈ aΓS ⊆ a ∪ aΓS.

If r = a t then r s = (a t) s = a (t s) ∈ aΓS⊆ a ∪ aΓS.

Therefore r s ∈ a ∪ aΓS and hence a ∪ aΓS is a right Γ-ideal of S.

Let R be a right Γ-ideal of S containing a.

Let r ∈ a ∪ aΓS. Then r = a or r = a t for some t ∈ S, α ∈ Γ.

If r = a then r = a ∈ R. If r = a t then r = a t ∈ R.

Therefore a ∪ aΓS ⊆ R and hence a ∪ SΓa is the smallest right Γ-ideal containing a.

Therefore R(a) = a∪aΓS.

NOTE 2.1.39 : If S is a Γ-semigroup and a ∈ S then R (a) = aΓS1.

We now introduce a principal Γ-ideal of a Γ-semigroup.

DEFINITION 2.1.40 : A Γ- ideal A of a Γ-semigroup S is said to be a principal Γ- ideal

provided A is a Γ- ideal generated by {a} for some a ∈ S. It is denoted by J[a] or <a>.

Now we characterize principal Γ-ideal of a Γ-semigroup.

THEOREM 2.1.41 : If S is a -semigroup and a ∈ S then

J(a) = a ∪ aΓS ∪ SΓa ∪ SΓaΓS.

Proof : Let s ∈ S, r ∈ a ∪ aΓS ∪ SΓa ∪ SΓaΓS and γ ∈ Γ.

r ∈ a ∪ aΓS ∪ SΓa ∪ SΓaΓS ⇒ r = a or r = aαt or r = tαa or r = tαa u for some

t, u ∈ S and α, ∈ Γ.

If r = a then rγs = aγs ∈ aΓS and sγr = sγa ∈ SΓa.

If r = aαt then rγs = (aαt)γs = aα(tγs) ∈ aΓS and sγr = sγ(aαt) = sγaαt ∈ SΓaΓS.

If r = tαa then rγs = (tαa)γs = tαaγs ∈ SΓaΓS or sγr = sγ(tαa) = (sγt)αa ∈ SΓa.

If r = tαa u then rγs = (tαa u)γs = tαa (uγs) ∈ SΓaΓS

and sγr = sγ(tαa u) = (sγt)αa u ∈ SΓaΓS.

But aΓS, SΓa, SΓaΓS are all subsets of aΓS ∪ SΓa ∪ SΓaΓS.

Therefore rγs, sγr ∈ a ∪ aΓS ∪ SΓa ∪ SΓaΓS and hence a ∪ aΓS ∪ SΓa ∪ SΓaΓS is a

Γ-ideal of S.

Let J be a Γ-ideal of S containing a. Let r ∈ a ∪ aΓS ∪ SΓa ∪ SΓaΓS.

Then r = a or r = aαt or r = tαa or r = tαa u for some t, u ∈ S and α, ∈ Γ.

Structure and ideal theory of duo Γ-semigroups Ideals in Γ-semigroups

66

If r = a then r = a ∈ J. If r = aαt then r = aαt ∈ J.

If r = tαa then r = tαa ∈ J. If r = tαa u then r = tαa u ∈ J.

Therefore a ∪ aΓS ∪ SΓa ∪ SΓaΓS ⊆ J.

Hence a ∪ aΓS ∪ SΓa ∪ SΓaΓS is the smallest Γ-ideal of S containing a.

Therefore J(a) = a ∪ aΓS ∪ SΓa ∪ SΓaΓS.

NOTE 2.1.42 : If S is a Γ-semigroup and a ∈ S, then <a> = a ∪ aΓS ∪ SΓa ∪ SΓaΓS

= S1Γa ΓS

1.

THEOREM 2.1.43 : In any -semigroup S, the following are equivalent.

(1) Principal -ideals of S form a chain.

(2) -ideals of S form a chain.

Proof : (1) ⇒ (2) : Suppose that principal Γ-ideals of S form a chain.

Let A, B be two Γ-ideals of S. Suppose if possible A ⊈ B, B ⊈ A.

Then there exists a ∈ A\B and b ∈ B\A.

a ∈ A ⇒ < a > ⊆ A and b ∈ B ⇒ < b > ⊆ B.

Since principal Γ-ideals form a chain, either < a > ⊆ < b > or < b > ⊆ < a >.

If < a > ⊆ < b >, then a ∈ < b > ⊆ B. It is a contradiction.

If < b > ⊆ < a >, then b ∈ < a > ⊆ A. It is also a contradiction.

Therefore either A ⊆ B or B ⊆ A and hence Γ-ideals from a chain.

(2) ⇒ (1) : Suppose that Γ-ideals of S form a chain.

Then clearly principal Γ-ideal of S form a chain.

THEOREM 2.1.44 : Let A and B be two Γ-ideals in a Γ-semigroup S.

Then Al (B) = { x ∈ S : < x >ΓB ⊆ A} is a Γ-ideal of S containing A.

Proof : Let x ∈ Al (B), s ∈ S and γ ∈ Γ.

x ∈ Al (B) < x >ΓB ⊆ A. Now < s γ x >ΓB ⊆ < x >ΓB ⊆ A s γ x ∈ A

l (B).

And < x γ s >ΓB ⊆ < x >ΓB ⊆ A x γ s ∈ Al (B).

Therefore s γ x, x γ s ∈ Al (B). Hence A

l (B) is a Γ-ideal of S containing A.

THEOREM 2.1.45 : Let A and B be two Γ-ideals in a Γ-semigroup S.

Then Ar (B) = { x ∈ S : B Γ< x > ⊆ A} is a Γ-ideal of S containing A.

Proof : Let x ∈ Ar (B), s ∈ S and γ ∈ Γ.

x ∈ Ar (B) BΓ< x > ⊆ A. Now BΓ< s γ x > ⊆ BΓ< x > ⊆ A s γ x ∈ A

r (B).

Structure and ideal theory of duo Γ-semigroups Ideals in Γ-semigroups

67

And BΓ< x γ s > ⊆ BΓ< x > ⊆ A x γ s ∈ Ar (B).

Therefore s γ x, x γ s ∈ Ar (B). Hence A

r (B) is a Γ-ideal of S containing A.

2.2. COMPLETELY PRIME -IDEALS AND PRIME -IDEALS

In this section, the terms; completely prime Γ-ideal, c-system, prime Γ-ideal,

m-system are introduced. It is proved that every -subsemigroup of a -semigroup is a

c-system. It is also proved that a -ideal P of a -semigroup S is completely prime if and

only if S\P is either a c-system or empty. It is proved that if P is a Γ- ideal of a

Γ-semigroup S, then the conditions (1) if A, B are Γ- ideals of S and AΓB⊆P then either

A⊆P or B⊆P, (2) if a, b ∈ S such that aΓS1Γb ⊆ P, then either a ∈ P or b ∈ P are

equivalent. It is also proved that every completely prime -ideal of a -semigroup S is a

prime -ideal of S. Further it is proved that a -ideal P of a -semigroup S is a prime

-ideal of S if and only if S\P is an m-system or empty. In a globally idempotent

Γ-semigroup, it is proved that every maximal Γ-ideal is a prime Γ-ideal. It is also proved

that a globally idempotent Γ-semigroup having a maximal Γ-ideal, contains semisimple

elements.

We now introduce the notion of a completely prime Γ-ideal of a Γ-semigroup

which is due to DHEENA and ELAVARASAN [14].

DEFINITION 2.2.1 : A Γ- ideal P of a Γ-semigroup S is said to be a completely prime

Γ- ideal provided x, y ∈ S and xΓy ⊆ P implies either x ∈ P or y ∈ P.

We now introduce the notion of a c-system of a Γ-semigroup.

DEFINITION 2.2.2 : Let S be a Γ-semigroup. A nonempty subset A of S is said to be a

c-system of S if for each a, b ∈ A there exists an element α ∈ Γ such that a b ∈ A.

THEOREM 2.2.3 : Every -subsemigroup of a -semigroup is a c-system.

Proof : Let T be a Γ-subsemigroup of S and a, b ∈ T, α ∈ Γ.

Since T is a Γ-subsemigroup of S, a b ∈ T for all α ∈ Γ. Therefore T is a c-system.

We now prove a necessary and sufficient condition for a -ideal to be a completely

prime -ideal in a Γ-semigroup.

Structure and ideal theory of duo Γ-semigroups Ideals in Γ-semigroups

68

THEOREM 2.2.4 : A -ideal P of a -semigroup S is completely prime if and only if

S\P is either a c-system of S or empty.

Proof : Suppose that P is a completely prime Γ-ideal of S and S\P ≠ ∅. Let a, b ∈ S\P.

Then a ∉ P, b ∉ P. Suppose if possible there exists no α ∈ Γ such that a b ∈ S\P.

Then aΓb ⊆ P. Since P is completely prime, either a ∈ P or b ∈ P. It is a contradiction.

Therefore there exists an α ∈ Γ such that a b ∈ S\P and hence S\P is a c-system.

Conversely suppose that S\P is a c-system of S or S\P is empty.

If S\P is empty then P = S and hence P is completely prime.

Assume that S\P is a c-system of S. Let a, b ∈ S and aΓb ⊆ P.

Suppose if possible a ∉ P and b ∉ P. Then a ∈ S\P and b ∈ S\P.

Since S\P is a c-system, there exists α ∈ Γ such that a b ∈ S\P.

Thus a b ∉ P and hence aΓb ⊈ P. It is a contradiction. Hence either a ∈ P or b ∈ P.

Therefore P is a completely prime Γ-ideal of S.

We now introduce the notion of a prime Γ-ideal of a Γ-semigroup which is due to

DHEENA and ELAVARASAN [14].

DEFINITION 2.2.5 : A Γ- ideal P of a Γ-semigroup S is said to be a prime Γ- ideal

provided A, B are two Γ-ideals of S and AΓB ⊆ P ⇒ either A ⊆ P or B⊆ P.

THEOREM 2.2.6 : If P is a Γ- ideal of a Γ-semigroup S, then the following conditions

are equivalent.

(1) If A, B are Γ- ideals of S and AΓB⊆P then either A⊆P or B⊆P.

(2) If a, b ∈ S such that aΓS1Γb ⊆ P, then either a ∈ P or b ∈ P.

Proof : (1) (2).

Suppose that if A, B are Γ- ideals of S and AΓB⊆P then either A⊆P or B⊆P.

Let Sba , and aΓS1Γb ⊆ P.

Now aΓS1Γb ⊆ P S

1ΓaΓS

1ΓbΓS

1 ⊆ P ⇒ < a > Γ < b > ⊆ P either < a > ⊆ P or < b >

⇒ either a ∈ P or b ∈ P.

(2) (1) : Suppose that if aΓS1Γb ⊆ P then either a ∈ P or b ∈ P.

Let A, B be two Γ-ideals of S and AΓB⊆P.

Suppose if possible A P and B P. Then there exists a ∈ A\P and b ∈ B\P.

Now aΓS1Γb ⊆ < a > Γ < b > ⊆ AΓB ⊆ P ⇒ either a ∈ P or b ∈ P.

It is a contradiction. Therefore either A⊆P or B⊆P.

Structure and ideal theory of duo Γ-semigroups Ideals in Γ-semigroups

69

COROLLARY 2.2.7 : A Γ- ideal P of a Γ-semigroup S is a prime Γ- ideal iff a, b ∈ S

such that aΓS1Γb ⊆ P, then either a ∈ P or b ∈ P.

THEOREM 2.2.8 : Every completely prime -ideal of a -semigroup S is a prime

-ideal of S.

Proof : Suppose that A is a completely prime -ideal of a -semigroup S.

Let a, b ∈ S and aΓS1Γb ⊆ A.

Now aΓb ⊆ aΓS1Γb ⊆ A. Since A is completely prime, either a ∈ A or b ∈ A.

Therefore A is a prime -ideal of S.

We now introduce the notion of an m-system of a Γ-semigroup.

DEFINITION 2.2.9 : A nonempty subset A of a Γ-semigroup S is said to be an m-system

provided for any a,b ∈ A , there exists an x ∈ S and α, β ∈ Γ such that aαx b ∈ A.

NOTE 2.2.10 : Every c-system of a Γ-semigroup S is an m-system of S.

We now prove a necessary and sufficient condition for a Γ-ideal to be a prime

Γ-ideal in a Γ-semigroup.

THEOREM 2.2.11 : A -ideal P of a -semigroup S is a prime -ideal of S if and only

if S\P is an m-system of S or empty.

Proof : Suppose that P is a prime Γ-ideal of a Γ-semigroup S and S\P ≠ ∅.

Let a, b ∈ S\P. Then a ∉ P, b ∉ P.

Suppose if possible there exist no x ∈ S and α, β ∈ Γ such that aαx b ∈ S\P.

Then aΓSΓb ⊆ P. Since P is prime, either a ∈ P or b ∈ P. It is a contradiction.

Therefore there exist an x ∈ S and some , ∈ Γ such that aαx b ∈ S\P.

Hence S\P is an m-system.

Conversely suppose that S\P is either an m-system of S or S\P = ∅.

If S\P is empty then P = S and hence P is prime. Assume that S\P is an m-system of S.

Let a, b ∈ S and aΓSΓb ⊆ P. Suppose if possible a ∉ P, b ∉ P. Then a, b ∈ S\P.

Since S\P is an m-system, there exists x ∈ S and , ∈ Γ such that aαx b ∈ S\P.

Thus aαx b ∉ P and hence aΓSΓb ⊈ P. It is a contradiction.

Therefore either a ∈ P or b ∈ P. Hence P is a prime Γ-ideal of S.

Structure and ideal theory of duo Γ-semigroups Ideals in Γ-semigroups

70

We now introduce the notion of a globally idempotent Γ-semigroup.

DEFINITION 2.2.12 : A Γ-semigroup S is said to be a globally idempotent Γ-semigroup

if SΓS = S.

THEOREM 2.2.13 : If S is a globally idempotent -semigroup then every maximal

-ideal of S is a prime -ideal of S.

Proof : Let M be a maximal Γ-ideal of S.

Let A, B be two Γ-ideals of S such that AΓB ⊆ M.

Suppose if possible A ⊈ M, B ⊈ M.

Now A ⊈ M ⇒ M ∪ A is a Γ-ideal of S and M ⊂ M ∪ A ⊆ S.

Since M is maximal, M ∪ A = S. Similarly B ⊈ M ⇒ M ∪ B = S.

Now S = SΓS = (M ∪ A)Γ(M ∪ B) = (MΓM) ∪ (MΓB) ∪ (AΓM) ∪ (AΓB) ⊆ M ⇒ S ⊆ M.

Thus M = S. It is a contradiction.

Therefore either A ⊆ M or B ⊆ M. Hence M is prime.

THEOREM 2.2.14 : If S is a globally idempotent -semigroup having maximal

-ideals then S contains semisimple elements.

Proof : Suppose that S is a globally idempotent Γ-semigroup having maximal Γ-ideals.

Let M be a maximal Γ-ideal of S. Then by theorem 2.2.13, M is prime.

Now if a ∈ S\M then < a >Γ< a > ⊈ M and hence S = M ∪ < a > = M ∪ (< a >Γ< a >).

Therefore a ∈ < a >Γ< a >. Thus a is semisimple.

Therefore S contains semisimple elements.

2.3. COMPLETELY SEMIPRIME -IDEALS AND SEMIPRIME -IDEALS

In this section, the terms; completely semiprime Γ-ideal, d-system, semiprime

Γ-ideal, n-system are introduced. It is proved that (1) every completely prime Γ-ideal of a

Γ-semigroup is completely semiprime (2) every completely semiprime Γ-ideal of a

Γ-semigroup is semiprime, (3) every prime Γ-ideal of a Γ-semigroup is a semiprime

Γ-ideal. It is also proved that the nonempty intersection of a family of (1) completely

prime Γ-ideals of a Γ-semigroup is completely semiprime, (2) prime Γ-ideals of a

Γ-semigroup is semiprime. It is proved that a -ideal P of a -semigroup S is completely

semiprime iff S\P is a d-system of S or empty. It is also proved that an ideal Q of a

Structure and ideal theory of duo Γ-semigroups Ideals in Γ-semigroups

71

Γ-semigroup S is semiprime iff S\Q is either an n-system or empty. Further it is proved

that if N is an n-system in a Γ-semigroup S and a ∈ N, then there exists an m-system M of

S such that a ∈ M and M ⊆ N.

We now introduce the notion of a completely semiprime Γ- ideal and a semiprime

Γ-ideal in a Γ-semigroup.

DEFINITION 2.3.1 : A Γ-ideal A of a Γ-semigroup S is said to be a completely

semiprime Γ- ideal provided xΓx ⊆ A ; x ∈S implies x ∈A.

THEOREM 2.3.2 : Every completely prime -ideal of a -semigroup S is a

completely semiprime -ideal of S.

Proof : Let A be a completely prime -ideal of a Γ-semigroup S.

Suppose that . and AxxSx

Since A is a completely prime -ideal of S, x ∈A.

Therefore S is completely semiprime -ideal.

THEOREM 2.3.3 : The nonempty intersection of any family completely prime

-ideals of a -semigroup S is a completely semiprime -ideal of S.

Proof : Let {Aα}α∈Δ be a family of completely prime Γ-ideals of S such that A ≠ ∅.

By theorem 2.1.17, A is a Γ-ideal.

Let a ∈ S, aΓa ⊆ A . Then aΓa ⊆ Aα for all α ∈ Δ.

Since Aα is completely prime, a ∈ Aα for all α ∈ Δ and hence a ∈ A .

Therefore A is a completely semiprime Γ-ideal of S.

We now introduce the notion of a d-system of a Γ-semigroup.

DEFINITION 2.3.4 : Let S be a Γ-semigroup. A nonempty subset A of S is said to be a

d-system of S if for each a ∈ A there exists an element α ∈ Γ such that a a ∈ A.

We now prove a necessary and sufficient condition for a Γ-ideal to be a completely

semiprime Γ-ideal in a Γ-semigroup.

THEOREM 2.3.5 : A -ideal P of a -semigroup S is completely semiprime iff S\P is

a d-system of S or empty.

Structure and ideal theory of duo Γ-semigroups Ideals in Γ-semigroups

72

Proof : Suppose that P is a completely semiprime Γ-ideal of S and S\P ≠ ∅.

Let a ∈ S\P. Then a ∉ P. Suppose if possible there exists no α ∈ Γ such that a a ∈ S\P.

Then aΓa ⊆ P. Since P is completely semiprime, a ∈ P. It is a contradiction.

Therefore there exists an α ∈ Γ such that a a ∈ S\P and hence S\P is a d-system of S.

Conversely suppose that S\P is a d-system of S or S\P is empty.

If S\P is empty then P = S and hence P is completely semiprime.

Assume that S\P is a d-system of S.

Let a ∈ S and aΓa ⊆ P. Suppose if possible a ∉ P. Then a ∈ S\P.

Since S\P is a d-system, there exists α ∈ Γ such that a a ∈ S\P.

Thus a a ∉ P and hence aΓa ⊈ P. It is a contradiction.

Hence a ∈ P. Therefore P is a completely semiprime Γ-ideal of S.

We now introduce the notion of a semiprime Γ-ideal of a Γ-semigroup..

DEFINITION 2.3.6 : A Γ- ideal A of a Γ-semigroup S is said to be a semiprime Γ- ideal

provided x ∈ S, xΓS1Γx ⊆ A implies x ∈ A.

THEOREM 2.3.7 : Every completely semiprime -ideal of a -semigroup S is a

semiprime -ideal of S.

Proof : Suppose that A is a completely semiprime -ideal of a -semigroup S.

Let a ∈ S and aΓS1Γa ⊆ A.

Now aΓa ⊆ aΓS1Γa ⊆ A. Since A is completely semiprime, a ∈ A.

Therefore A is a semiprime -ideal of S.

THEOREM 2.3.8 : Every prime -ideal of a -semigroup S is a semiprime -ideal

of S.

Proof : Suppose that A is a prime -ideal of a -semigroup S.

Let a ∈ S and aΓS1Γa ⊆ A. By corollary 2.2.7, a ∈ A.

Therefore A is a semiprime -ideal of S.

THEOREM 2.3.9 : The nonempty intersection of any family of prime -ideals of a

-semigroup S is a semiprime -ideal of S.

Proof : Let {Aα}α∈Δ be a family of prime Γ-ideals of S such that A ≠ ∅.

Structure and ideal theory of duo Γ-semigroups Ideals in Γ-semigroups

73

By theorem 2.1.17, A is a Γ-ideal.

Let a ∈ S, aΓSΓa ⊆ A . Then aΓSΓa ⊆ Aα for all α ∈ Δ.

Since Aα is prime, a ∈ Aα for all α ∈ Δ and hence a ∈ A .

Therefore A is a semiprime Γ-ideal of S.

We now introduce the notion of an n-system of a Γ-semigroup.

DEFINITION 2.3.10 : A nonempty subset A of a Γ-semigroup S is said to be an n-system

provided for any a ∈ A , there exists an x ∈ S and some α, β ∈ Γ such that a x a ∈ A.

NOTE 2.3.11 : Every d-system of a Γ-semigroup S is an n-system of S.

We now prove a necessary and sufficient condition for a Γ-ideal to be a semiprime

Γ-ideal in a Γ-semigroup.

THEOREM 2.3.12 : A Γ- ideal Q of a Γ-semigroup S is a semiprime Γ- ideal iff S\Q

is an n-system of S or empty.

Proof : Suppose that Q is a semiprime Γ-ideal of a Γ-semigroup S and S\Q ≠ ∅.

Let a ∈ S\Q. Then a ∉ Q.

Suppose if possible there exist no x ∈ S and , β ∈ Γ such that a x a ∈ S\Q.

Then aΓSΓa ⊆ Q. Since Q is semiprime, a ∈ Q. It is a contradiction.

Therefore there exists an x ∈ S and , ∈ Γ such that a x a ∈ S\Q.

Hence S\Q is an n-system.

Conversely suppose that S\Q is an n-system of S or S\Q = ∅.

If S\Q is empty then Q = S and hence Q is semiprime.

Assume that S\Q is an n-system of S.

Let a ∈ S and aΓSΓa ⊆ Q. Suppose if possible a ∉ Q. Then a ∈ S\Q.

Since S\Q is an n-system, there exists an x ∈ S and , ∈ Γ such that a x a ∈ S\Q.

Thus a x a ∉ Q and hence aΓSΓa ⊈ Q. It is a contradiction.

Hence a ∈ Q. Therefore Q is a semiprime Γ-ideal of S.

THEOREM 2.3.13 : If N is an n-system in a Γ-semigroup S and a ∈ N, then there

exists an m-system M in S such that a ∈ M and M ⊆N.

Structure and ideal theory of duo Γ-semigroups Ideals in Γ-semigroups

74

Proof : We construct a subset M of N as follows. Define a1= a.

Since a1∈ N and N is an n-system, a1αxβa1 ∈ N, for some x ∈ S, , β ∈ Γ.

Thus (a1ΓSΓa1)∩N ≠ ∅. Let a2 ∈ (a1ΓSΓa1)∩N.

Since a2 ∈ N and N is an n-system, (a2ΓSΓa2)∩N ≠ ∅ and so on.

In general, if ai has been defined with ai ∈ N, choose ai+1 as an element of (aiΓSΓai)∩N.

Let M = { a1, a2,……, ai, ai+1 …….}. Now a ∈ M and M ⊆ N.

We now show that M is an m-system.

Let ai, aj ∈ M (for i≤ j). Then aj+1 ∈ ajΓSΓaj ⊆ aiΓSΓai ⇒ aj+1 = ai x aj, x ∈ S, , β ∈ Γ.

But aj+1 ∈ M, so aj+1 = ai x aj ∈ M, for x ∈ S , β ∈ Γ. Therefore M is an m-system.

2.4. PRIME -RADICAL AND COMPLETELY PRIME -RADICAL

In this section, the terms; prime Γ-radical rad A ( A ), completely prime Γ-radical

c.rad A of a Γ-ideal A in a Γ-semigroup are introduced. If A and B are two Γ-ideals of a

Γ-semigroup S, then it is proved that (i) A ⊆ B ⇒ √(A) ⊆ √(B) and c.rad A ⊆ c.rad B,

(ii) √(AΓB) =√(A∩B) =√(A) ∩ √(B) and c.rad (AΓB) = c.rad (A∩B) = c.rad (A)

∩ c.rad (B), (iii) √(√(A)) = √(A) and c.rad (c.rad A) = c.rad A. If A is a Γ-ideal in a

Γ-semigroup S then it is proved that rad A is a semiprime Γ-ideal and c.rad A is a

completely semiprime Γ-ideal of S. It is proved that a Γ- ideal Q of Γ-semigroup S is a

semiprime Γ- ideal of S iff √(Q) = Q. It is also proved that if A is a Γ- ideal of a

Γ-semigroup S, then √(A) is the smallest semiprime Γ- ideal of S containig A. It is proved

that if P is a prime -ideal of a -semigroup S, then √((P )n-1

P) = P for all n ∈ N. It is

proved that in a -semigroup S with identity there is a unique maximal -ideal M such that

√((M )n-1

M) = M for all n ∈ N. Further it is proved that if A is a -ideal of a -semigroup

S then √A = {x ∈ S: every m-system of S containing x meets A} i.e, √A = {x ∈ S : M(x) ∩

A ≠ ∅}.

We now introduce prime Γ-radical and complete prime Γ-radical of a Γ-ideal in a

Γ-semigroup.

DEFINITION 2.4.1 : If A is a Γ-ideal of a Γ-semigroup S, then the intersection of all

prime Γ-ideals of S containing A is called prime Γ-radical or simply Γ-radical of A and it

is denoted by √A or rad A.

Structure and ideal theory of duo Γ-semigroups Ideals in Γ-semigroups

75

DEFINITION 2.4.2 : If A is a Γ-ideal of a Γ-semigroup S, then the intersection of all

completely prime Γ-ideals of S containing A is called complete prime Γ-radical or simply

complete Γ-radical of A and it is denoted by c. rad A.

THEOREM 2.4.3 : If A and B are any two Γ-ideals of a Γ-semigroup S, then

(i) A ⊆ B ⇒ √(A) ⊆ √(B) .

(ii) √(AΓB) = √(A∩B) = √(A) ∩ √(B).

(iii) √(√(A)) = √(A).

Proof : (i) Suppose that A ⊆ B. If P is a prime Γ-ideal containing B then P is a prime

Γ-ideal containing A. Therefore √A ⊆√ B.

(ii) Let P be a prime Γ-ideal containing AΓB. Then AΓB ⊆ P ⇒ A ⊆ P or B ⊆ P

⇒ A ∩ B ⊆ P. Therefore P is a prime Γ-ideal containing A ∩ B.

Therefore √(A ∩ B) ⊆ √(AΓB). Now let P be a prime Γ-ideal containing A ∩ B.

Since A is a Γ-ideal of S, AΓB ⊆ AΓS⊆A. Since B is a Γ-ideal of S, AΓB ⊆ SΓB⊆ B.

Therefore AΓB ⊆ A ∩ B ⊆ P ⇒ AΓB ⊆ P.

Hence P is a prime Γ-ideal containing AΓB. Therefore √(AΓB) ⊆ √(A ∩ B).

Therefore √(AΓB) = √(A ∩ B). Now AΓB ⊆ A, AΓB ⊆ B ⇒ √(AΓB) ⊆ √A,

√(AΓB) ⊆ √B, by condition (i). Hence √(AΓB) ⊆ √A ∩ √B.

Let x ∈ √A ∩ √B. Then x ∈ √A and x ∈ √B.

Suppose if possible x ∉ √(AΓB). Then there exists a prime Γ-ideal P containing (AΓB)

and not containing x. AΓB ⊆ P ⇒ either A ⊆ P or B ⊆ P.

If A ⊆ P then P is a prime Γ-ideal containing A and not containing x ⇒ x ∉ √A.

If B ⊆ P then P is a prime Γ-ideal containing B and not containing x ⇒ x ∉ √B.

It is a contradiction. Therefore x ∈ √(AΓB). Therefore √A ∩ √B ⊆ √(AΓB).

Therefore √A ∩ √B = √(AΓB). Hence √(AΓB) = √(A∩B) = √(A) ∩ √(B).

(iii) √A = The intersection of all prime Γ-ideals of S containing A.

Now √(√A) = The intersection of all prime Γ-ideals of S containing √A.

= The intersection of all prime Γ-ideals of S containing A = √A.

Therefore √(√A) = √A.

THEOREM 2.4.4 : If A and B are any two Γ-ideals of a Γ-semigroup S, then

(i) A ⊆ B ⇒ c.rad A ⊆ c.rad B.

(ii) c.rad (AΓB) = c.rad (A∩B) = c.rad (A) ∩ c.rad (B).

(iii) c.rad (c.rad A) = c.rad A.

Structure and ideal theory of duo Γ-semigroups Ideals in Γ-semigroups

76

Proof : (i) Suppose that A ⊆ B. If P is a completely prime Γ-ideal containing B then P is a

completely prime Γ-ideal containing A. Therefore c.rad A ⊆ c.rad B.

(ii) Let P be a completely prime Γ-ideal containing AΓB.

Then AΓB ⊆ P ⇒ A ⊆ P or B ⊆ P ⇒ A ∩ B ⊆ P.

Therefore P is a completely prime Γ-ideal containing A ∩ B.

Therefore c. rad (A ∩ B) ⊆ c. rad (AΓB).

Let P be a completely prime Γ-ideal containing A ∩ B.

Since A is a Γ-ideal of S, AΓB ⊆ AΓS ⊆ A. Since B is a Γ-ideal of S, AΓB ⊆ SΓB ⊆ B.

Therefore AΓB ⊆ A ∩ B ⊆ P ⇒ AΓB ⊆ P.

Hence P is a completely prime Γ-ideal containing AΓB.

Therefore c. rad (AΓB) ⊆ c. rad (A ∩ B). Hence c. rad (AΓB) = c. rad (A ∩ B).

Now AΓB ⊆ A, AΓB ⊆ B ⇒ c. rad (AΓB) ⊆ c. rad A, c. rad (AΓB) ⊆ c. rad B,

by condition (i). Hence c.rad (AΓB) ⊆ c. rad A ∩ c. rad B.

Let x ∈ c. rad A ∩ c. rad B. Then x ∈ c. rad A and x ∈ c. rad B.

Suppose if possible x ∉ c. rad (AΓB).

Then there exists a completely prime Γ-ideal P containing (AΓB) and not containing x.

AΓB ⊆ P ⇒ either A ⊆ P or B ⊆ P.

If A ⊆ P then P is a completely prime Γ-ideal containing A and not containing x

and hence x ∉ c. rad A.

If B ⊆ P then P is a completely prime Γ-ideal containing B and not containing x

and hence x ∉ c. rad B. It is a contradiction. Therefore x ∈ c. rad (AΓB).

Therefore c. rad A ∩ c. rad B ⊆ c. rad (AΓB).

Therefore c. rad A ∩ c. rad B = c. rad (AΓB).

Hence c.rad (AΓB) = c.rad (A∩B) = c.rad (A) ∩ c.rad (B).

(iii) c. rad A = The intersection of all prime Γ-ideals of S containing A.

Now c. rad (c. rad A) = The intersection of all prime Γ-ideals of S containing c. rad A.

= The intersection of all prime Γ-ideals of S containing A = c. rad A.

Therefore c. rad (c. rad A) = c. rad A.

THEOREM 2.4.5 : If A is a -ideal of a -semigroup S then √A is a semiprime -ideal

of S.

Proof : By theorem 2.3.9, √A is a semiprime Γ-ideal of S.

Structure and ideal theory of duo Γ-semigroups Ideals in Γ-semigroups

77

THEOREM 2.4.6 : A Γ- ideal Q of Γ-semigroup S is a semiprime Γ- ideal of S iff

√(Q) = Q .

Proof : Suppose that Q is a semiprime Γ-ideal of S. Clearly Q ⊆ √Q.

Since Q is a semiprime Γ-ideal of S contains Q, √Q ⊆ Q. Thus √Q = Q.

Conversely Suppose that Q is a Γ-ideal of S such that √Q = Q.

By theorem 2.4.5, √Q is a semiprime Γ-ideal of S.

Therefore Q is a semiprime Γ-ideal of S.

COROLLARY 2.4.7 : If A is a Γ- ideal of a Γ-semigroup S, then √(A) is the smallest

semiprime Γ- ideal of S containig A.

Proof : By theorem 2.4.5, √(A) is a semiprime Γ-ideal of S. Clearly A ⊆ √A.

Let Q be any semiprime Γ- ideal of S containing A.

By theorem 2.4.3, A ⊆ Q ⇒ √(A) ⊆ √(Q).

Since Q is semiprime, by theorem 2.4.6, √(Q) = Q. Therefore, √(A) ⊆ Q.

Hence √(A) is the smallest semiprime Γ- ideal of S containing A.

THEOREM 2.4.8 : If A is a -ideal of a -semigroup S then c.rad A is a completely

semiprime -ideal of S.

Proof : By theorem 2.3.3, c.rad A is a completely semiprime Γ-ideal of S.

THEOREM 2.4.9 : A Γ- ideal Q of Γ-semigroup S is a completely semiprime Γ- ideal

of S iff c. rad (Q) = Q .

Proof : Suppose that Q is a completely semiprime Γ-ideal of S. Clearly Q ⊆ c. rad Q.

Since Q is a completely semiprime Γ-ideal of S contains Q, c. rad Q ⊆ Q.

Thus c. rad Q = Q.

Conversely Suppose that Q is a Γ-ideal of S such that c. rad Q = Q.

By theorem 2.4.8, c. rad Q is a completely semiprime Γ-ideal of S.

Therefore Q is a completely semiprime Γ-ideal of S.

COROLLARY 2.4.10 : If A is a Γ- ideal of a Γ-semigroup S, then c. rad (A) is the

smallest semiprime Γ- ideal of S containig A.

Proof : By theorem 2.4.9, c. rad (A) is a completely semiprime Γ-ideal of S.

Clearly A ⊆ c. rad A.

Structure and ideal theory of duo Γ-semigroups Ideals in Γ-semigroups

78

Let Q be any completely semiprime Γ- ideal of S containing A.

By theorem 2.4.4, A ⊆ Q ⇒ c. rad (A) ⊆ c. rad (Q).

Since Q is completely semiprime, by theorem 2.4.9, c. rad (Q) = Q.

Therefore, c. rad (A) ⊆ Q.

Hence c. rad (A) is the smallest completely semiprime Γ-ideal of S containing A.

THEOREM 2.4.11 : If P is a prime -ideal of a -semigroup S, then √((P )n-1

P) = P

for all n ∈ N.

Proof : We use induction on n, to prove √((P )n-1

P) = P. First we prove that √P = P.

Since P is a prime Γ-ideal, P ⊆ √P ⊆ P ⇒ √P = P.

Assume that √((PΓ)k-1

P) = P for k ∈ N such that 1 ≤ k < n.

Now √((PΓ)kP) = √((PΓ)

k-1PΓP) = √((PΓ)

k-1P∩ P) = √((PΓ)

n-1P)∩√P=√P ∩ √P=P∩P=P.

Therefore √((PΓ)kP) = P. By induction √((P )

n-1P) = P for all n ∈ N.

THEOREM 2.4.12 : In a -semigroup S with identity there is a unique maximal

-ideal M such that √((M )n-1

M = M for all n ∈ N.

Proof : Since S contains identity, S is a globally idempotent Γ-semigroup.

Since M is a maximal Γ-ideal of S, by theorem 2.2.13, M is prime.

By theorem 2.4.11, √((M )n-1

M = M for all n ∈ N.

THEOREM 2.4.13 : If A is a -ideal of a -semigroup S then √A = {x ∈ S : every

m-system of S containing x meets A} i.e, √A = { x ∈ S : M(x) ∩ A ≠ ∅}.

Proof : Suppose that x ∈ √A. Let M be an m-system containing x.

Then S\M is a prime Γ-ideal of S and x ∉ S\M.

If M ∩ A = ∅ then A ⊆ S\M.

Since S\M is a prime Γ-ideal containing A, √A ⊆ S\M and hence x ∈ S\M.

It is a contradiction. Therefore M(x) ∩ A ≠ ∅. Hence x ∈ {x ∈ S : M(x) ∩ A ≠ ∅}.

Conversely suppose that x ∈ { x ∈ S : M(x) ∩ A ≠ ∅}.

Suppose if possible x ∉ √A. Then there exists a prime Γ-ideal P such that x ∉ P.

Now S\P is an m-system and x ∈ S\P.

Therefore A ⊆ P ⇒ S\P ∩ A = ∅ ⇒ x ∉ {x ∈ S : M(x) ∩ A ≠ ∅}.

It is a contradiction. Therefore x ∈ √A. Thus √A = { x ∈ S : M(x) ∩ A ≠ ∅}.

* * * * *