**InternationaleJournals**

**Primary r- ideals in duo r-Semigroups.**

**A.
Gangadhara Rao**^{1}, A. Anjaneyulu^{2 }and D. Madhusudhana Rao^{3}

Dept. of Mathematics, V S R & N V R College,
Tenali, A.P. India. Emails: ^{1}raoag1967@gmail.com, ^{2 }anjaneyulu.addala@gmail.com, ^{3
}dmrmaths@gmail.com

**Abstract**

In this paper the terms left duo
Γ- semigroup,
right duo Γ- semigroup, duo Γ- semigroup are
introduced. It is proved that
a Γ-semigroup S is a duo Γ- semigroup if
and only if

*x*횪;S^{1}= S^{1}횪;*x *for all *x *∈ S. Further it is proved that every quasi
commutative Γ-semigroup is a duo Γ-semigroup. If A is a Γ-ideal in a (Left or
right) duo 횪;-semigroup S, then it is proved that *x*,* y *∈* S*, *x*Γ*y *⊆* A
x*Γ*s*Γ*y *⊆*A*. If A is a Γ-ideal in a duo Γ-semigroup S, then it is proved that *A*_{r}(*a*)* = *{*
x *∈* S
: a*횪;*x *⊆
A* *} is a 횪;-ideal of S for all *a *∈ S and *A*_{l}(*a*)* = *{*
x *∈* S
: x*횪;*a *⊆
*A *} is a

횪;-ideal of S for all *a *∈* S*. It is proved that if A is a Γ-ideal in a duo 횪;-semigroup S, then

(1) *a*횪;*b *⊆ A* *if and only if* < a*_{ }> 횪;* < b > *⊆ A and (2) *a*_{1}횪;*a*_{2}횪;…..*a*_{n}_{-1}횪;*a*_{n} ⊆
A if and only if

<*a*_{1}> 횪; <*a*_{2}>….횪; < *a*_{n}
> ⊆ A.** **Further it is proved that if A is a
Γ-ideal in a duo 횪;-semigroup S then for any natural number n, (* a*
)^{n-1}a
⊆ *A
*implies (*< a > *횪;* *)^{n-1} < a > ⊆* A.* If A_{1}
= the intersection of all completely prime Γ-ideals of S containing A, A_{2}
= {*x* ∈ S **:** (*x*Γ)^{n-1}x ⊆
A for some natural number *n* }, A_{3}
= the intersection of all prime ideals of S containing A,

A_{4} = {*x* ∈
S **:** (< *x*>Γ)^{n-1}^{
}<*x*> ⊆
A for some natural number *n* } for a Γ-ideal A of a
Γ-semigroup S, then it is proved that A_{2} is the minimal completely semiprime Γ-ideal of S containing A, A_{4 } is the minimal semiprime Γ-ideal of S containing A and A_{1} = A_{2} = A_{3}
= A_{4}.
It is proved that if A is a 횪;-ideal of a duo 횪;-semigroup S, then (1) A is completely prime if
and only if A is a prime and (2) A is a completely semiprime if and only if A
is a semiprime. If S is a duo 횪;-semigroup, then it is proved
that (1) S is strongly archimedean if and only if archimedean and (2) S is archimedean if and only if S has no proper prime 횪;-ideals.
Further it is proved that if S is a duo 횪;-semigroup, then the
conditions (1) S is strongly Archimedean, (2) S is Archimedean and (3) S has no
proper prime 횪;-ideals are equivalent.

**SUBJECT CLASSIFICATION (2010) : **20M07,
20M11, 20M12.

**KEY WORDS :**** ** left duo Γ-semigroup
right
duo Γ-semigroup and duo Γ-semigroup.

**International
eJournal of Mathematics and Engineering**

**Volume 3,
Issue ****3,** **Pages: 1642 - 1653**