**InternationaleJournals**

**Primary and
semiprimary ****г****-ideals In ****г**** ****-semigroups**

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

Department of Mathematics, V S R & N V R College, Tenali, A.P. India. Email ID: dmrmaths@gmail.com , anjaneyulu.addala@gmail.com , raoag1967@gmail.com

**ABSTRACT **

In
this paper a Γ -semigroup S
with identity, if √A = M for some Γ -ideal of S, where M is the unique
maximal Γ-ideal of S,
then A is a primary Γ -ideal. It is proved that if S is a Γ -semigroup with
identity, then for any natural number *n*,
(MΓ)^{n-1}M
is primary Γ -ideal of S,
where M is the unique maximal Γ-ideal
of S. Further it is proved that in quasi commutative Γ-semigroup S, a Γ-ideal A of S is
left primary iff A is right primary. It is proved that in semipseudo symmetric semiprimary
Γ-semigroup, globally idempotent principal Γ-ideals form a chain under set
inclusion. In a semisimple Γ-semigroup S
it is proved that the conditions every ideal of S is prime, S is a primary Γ-semigroup,
S is a left primary Γ-semigroup, S is a right primary Γ-semigroup, S is a
semiprimary Γ-semigroup, prime Γ-ideals of S form a chain, principal Γ-ideals
of S form a chain, Γ-ideals of S form a chain are equivalent.

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

**Key words**: Left Primary
Γ-ideal, right primary Γ-ideal, primary Γ-ideal, left primary Γ-semigroup,
right primary Γ-semigroup and primary Γ-semigroup, semiprimary Γ-ideal.

**International
eJournal of Mathematical Sciences, Technology and Humanities**

**Volume** 2,
**Issue** 1, **Pages**: **282 - 293**