Primary and semiprimary г-ideals In г -semigroups
D. Madhusudhana Rao1, A. Anjaneyulu2, A. Gangadhara Rao3.
Department of Mathematics, V S R & N V R College, Tenali, A.P. India. Email ID: firstname.lastname@example.org , email@example.com , firstname.lastname@example.org
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-1M 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