InternationaleJournals

Primary r- ideals in duo r-Semigroups.

Dept. of Mathematics, V S R & N V R College, Tenali, A.P. India. Emails: 1raoag1967@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;S1= S1횪;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,  xΓy A xΓsΓy A.  If A is a Γ-ideal  in a duo Γ-semigroup S, then  it is proved that Ar(a) = { x S : a횪;x A } is a 횪;-ideal of S for all a S and Al(a) = { x S : x횪;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) a1횪;a2횪;…..an-1횪;an A if and only if
<a1>
횪; <a2>….횪; < an > 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-1a A implies (< a > 횪; )n-1 < a > A.  If A1 = the intersection of all completely prime Γ-ideals of S containing A, A2 = {x S : (xΓ)n-1x A for some natural number n }, A3 = the intersection of all prime ideals of S containing A,
A4 = {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 A2 is the minimal completely semiprime Γ-ideal  of S containing A,  A4  is the minimal semiprime Γ-ideal  of S containing A and  A1 = A2 = A3 = A4.  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