**InternationaleJournals**

**Duo Noetherian ****-Semigroups**

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

Dept. of Mathematics, V S R & N V R College,
Tenali, A.P. India.

^{1}raoag1967@gmail.com, ^{2}anjaneyulu.addala@gmail.com, ^{3}dmrmaths@gmail.com

**ABSTRACT**

In
this paper the terms noetherian Γ-semigroup, Γ-closed Γ-semigroup
and center of a

Γ-semigroup are introduced. It is proved that if S is a noetherian Γ-semigroup containing proper Γ-ideals,
then S has a maximal Γ-ideal. It is
proved that if
H is the collection of all

Γ-ideals
in a Γ-closed duo Γ-semigroup S which are not
principal and H ≠ ∅, then there exists a prime Γ-ideal of S which
is not a principal Γ-ideal. It is proved
that if every
prime Γ-ideal
including S is principal in a Γ-closed duo Γ-semigroup S, then every
Γ-ideal in S is principal. It is proved
that if S is
a Γ-closed duo Γ-semigroup, which is a union of finite number
of principal Γ-ideals and every proper prime Γ-ideal is principal, then every
Γ-ideal is an intersection of a
principal Γ-ideal and an S-Primary Γ-ideal.
Also it is proved that if
S is a Γ-closed duo

Γ-semigroup, which is a union of finite number of principal Γ-ideals and every
proper prime

Γ-ideal of S is principal and S = S ΓS then every proper Γ-ideal is principal.** **If S is a duo

Γ-semigroup such that S and
every maximal Γ-ideal
is principal then it is proved that

(1) S has at most two maximal Γ-ideals and (2) if P is a proper prime Γ-ideal
of S then either P is a principal Γ-ideal or P = *x* ΓP for some .** **If every maximal Γ-ideal in a Γ-closed
duo

Γ-semigroup S is principal and S , for
every , then it is proved that S is a union of two
principal Γ-ideals
and every Γ-ideal is an intersection of a prime Γ-ideal and an

S-primary Γ-ideal. If S is a noetherian
or archimedian duo Γ-semigroup
such that S = and suppose for all , which is not a product of power of , then it
is proved that S is finitely generated and in particular if S is noetherian
strongly cancellative Γ-semigroup
without identity then S is finitely generated.
If S is a
duo Γ-semigroup
which is a union of finite number of principal Γ-ideals and if S =, then it is proved that S contains Γ-idempotent elements. If S is a strongly Γ-cancellable duo Γ-semigroup which is a union
of finite number of principal Γ-ideals, then it is proved that S contains
identity if and only if S =. In an archimedian duo Γ-semigroup S, if S is a
union of finite number of principal Γ-ideals or S contains a maximal Γ-ideal
which is finitely generated, then it is
proved that every proper

Γ-ideal is principal and S is a union of at most two principal Γ-ideals. It is proved that if A is a finitely
generated Γ-ideal
of a duo
Γ-semigroup
S, A = A ΓB for some Γ-ideal B and then for some . If S be a duo Γ-semigroup containing no
Γ-idempotents except perhaps the identity 1 and P is a finitely generated prime
Γ-ideal contained properly in for some
and , then
it is proved that (1) P does not contain any strongly

Γ-cancellable element and (2)
if A is finitely generated Γ-ideal containing a strongly

Γ-cancellable element then A for any
proper Γ-ideal
B. It is proved that if A is a
finitely generated Γ-ideal
of a duo Γ-semigroup S and A^{w}
= B such that A ΓB = where are
primary Γ-ideals,
then A ΓB = B. If S is a noetherian duo Γ-semigroup without
Γ-idempotents except perhaps identity, then
it is proved that for any Γ-ideal A, A^{w}^{ } where Z is the set of all non-cancellable
elements and A^{w}
= if S is strongly
Γ-cancellative. If S is a noetherian Γ-closed
duo Γ-monoid with a unique maximal
Γ-ideal M
= < *m* > for some *m* and if *x* then it
is proved that , *u*
is a unit or with Γx Γs. If S is a noetherian duo

Γ-monoid with a unique maximal
Γ-ideal M
= < *m* > for some *m* and if P
is a proper prime Γ-ideal
of S such that P M, then it
is proved that P . If S
is a noetherian duo Γ-monoid with a unique maximal
Γ-ideal M
= < *m* > for some *m* and if S
has no Γ-idempotents
except 1, then it is proved that is a prime Γ-ideal and also if Z M where
Z is the set of all non cancellable elements of S, then Z = If T is
a Γ-closed duo Γ-semigroup and S is a duo

Γ-semigroup such that S is a
Γ-subsemigroup of T and T = *x* ΓS^{1}
for some *x* and if S
is noetherian then it is proved that T is noetherian. Further an analogue of Hilbert basis theorem has obtained for duo Γ-semigroups.

**Mathematical subject classification (2010)** : 20M07; 20M11; 20M12.

__KEY WORDS__ : chained Γ-semigroup,
duo chained Γ-semigroup, noetherian
Γ-semigroup and center of a Γ-semigroup.

**International
eJournal of Mathematics and Engineering**

**Volume 3,
Issue ****3,** **Pages: 1688 - 1704**