Abstract

In this paper, the terms, Maximal Γ-ideal, primary Γ-semigroup and Prime Γ-ideal are introduced. It is proved that if S is a Γ- semigroup with identity and if ( non zero, assume this if S has zero) proper prime Γ-ideals in S are maximal then S is primary Γ-semigroup. Also it is proved that if S is a right cancellative quasi commutative Γ-emigroup and if S is a primary Γ- semigroup or a Γ- semigroup in which semiprimary Γ- ideals are primary, then for any primary Γ-ideal Q, √ Q is non-maximal implies Q = √ Q is prime. It is proved that if S is a right cancellative quasi commutative Γ-semigroup with identity, then 1) Proper prime Γ-ideals in S are maximal. 2) S is a primary Γ-semigroup. 3) Semiprimary Γ-ideals in S are primary, 4) If x and y are not units in S, then there exists natural numbers n and m such that (x Γ)n-1 x = yΓs and (yΓ) m-1 y = xΓr. For some s, r ∈ S are equivalent. Also it is proved that if S is a duo Γ–semigroup with identity, then 1) Proper prime Γ– ideals in S are maximal. 2) S is either a Γ– group and so Archimedian or S has a unique prime Γ–ideal P such that S = G∪ P, where G is the Γ–group of units in S and P is an Archimedian sub Γ–semi group of S are equivalent. In either case S is a primary Γ–semigroup and S has atmost one idempotent different from identity. It is proved that if S is a duo Γ-semigroup without identity, then S is a primary Γ-semigroup in which proper prime Γ-ideals are maximal if and only if S is an Archimedian Γ-semigroup. It is also proved that if S is a quasi commutative Γ-semigroup containing cancellable elements, then 1) The proper prime Γ-ideals in S are maximal. 2) S is a Γ-group or S is a cancellative Archimedian Γ-semigroup not containing identity or S is an extension of an Archimedian Γ-semigroup by a Γ-group S containing an identity are equivalent.

Keywords

References

[1] Anjaneyulu. A, and Ramakotaiah. D., On a class of semigroups, Simon stevin, Vol.54(1980), 241-249.
[2] Anjaneyulu. A., Structure and ideal theory of Duo semigroups, Semigroup Forum,Vol.22(1981), 257-276.
[3] Anjaneyulu. A., Semigroup in which Prime Ideals are maximal,Semigroup Forum,Vol.22(1981), 151-158.
[4] Giri. R. D. and Wazalwar. A. K., Prime ideals and prime radicals in non-commutative semigroup, Kyungpook Mathematical Journal Vol.33(1993), no.1,37-48.
[5] Madhusudhana Rao.D, Anjaneyulu. A and GangadharaRao. A, Prime Γ-radicals in Γ-semigroups, International e-Journal of Mathematics and Engineering138(2011) 1250 - 1259.
[6] Madhusudhana Rao. D, Anjaneyulu. A and GangadharaRao. A, Semipseudo symmetric Γ-idealsinΓ-semigroups,, International Journal of Mathematical Sciences, Technology and Humanities 18 (2011) 183-192.
[7] Madhusudhana Rao. D, Anjaneyulu. A and Gangadhara Rao. A, Primary and Semiprimary Γ-ideals in Γ-semigroup, International Journal of Mathematical Sciences, Technology and Humanities 29 (2012) 282-293.
[8] MadhusudhanaRao. D,Anjaneyulu. A and GangadharaRao. A, Pseudo symmetric Γ-ideals in Γ-semigroups, International e-Journal of Mathematics and Engineering 116(2011) 1074-1081.
[9] Petrich. M., Introduction to semigroups, Merril Publishing Company, Columbus,Ohio,(973).
[10] Sen. M.K. and Saha. N.K., On Γ -Semigroups-I, Bull. Calcutta Math. Soc. 78(1986),No.3, 180-186.
[11] Sen. M.K. and Saha. N.K., On Γ - Semigroups-II, Bull. Calcutta Math. Soc. 79(1987),No.6, 331-335
[12] Sen. M.K. and Saha. N.K., On Γ - Semigroups-III, Bull. Calcutta Math. Soc. 80(1988), No.1, 1-12.