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Duo Noetherian -Semigroups

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

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.   I
f 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 Aw = 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, Aw  where Z is the set of all non-cancellable elements and Aw =  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 ΓS1 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