Γ-Semigroups in which Prime Γ-Ideals are Maximal

S. Savithri; A. Gangadhara Rao; A. Anjaneyulu; J. M. Pradeep

doi:https://doi.org/10.14445/22315373/IJMTT-V49P547

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 49 | Number 5 | Year 2017 | Article Id. IJMTT-V49P547 | DOI : https://doi.org/10.14445/22315373/IJMTT-V49P547

Γ-Semigroups in which Prime Γ-Ideals are Maximal

S. Savithri, A. Gangadhara Rao, A. Anjaneyulu, J. M. Pradeep

Citation :

S. Savithri, A. Gangadhara Rao, A. Anjaneyulu, J. M. Pradeep, "Γ-Semigroups in which Prime Γ-Ideals are Maximal," International Journal of Mathematics Trends and Technology (IJMTT), vol. 49, no. 5, pp. 298-306, 2017. Crossref, https://doi.org/10.14445/22315373/IJMTT-V49P547

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

Γ-semigroup, Maximal Γ-ideal, primary Γ-semigroup, commutative Γ-semigroup, left (right) identity, identity, Zero element and Prime Γ-ideal.

References

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