On Weak Γ Multiplication and Weak Γ Comultiplication SΓ-Acts Over Commutative Γ-Semigroups

Samer Adnan  Jubair; Maryam Sabbeh  Al-Rubaiea

Authors

Samer Adnan Jubair Department of Mathematics, College of Education, Al-Qasim Green University, IRAQ
Maryam Sabbeh Al-Rubaiea Department of Mathematics, College of Education, Al-Qasim Green University, IRAQ

Keywords:

Weak Γ multiplication, Γ comultiplication, SГ-act, cyclic SГ-act, Γ-artinian Γ semigroup

Abstract

This paper introduces and investigates weak Γ multiplication SГ-acts and weak Γ comultiplication SГ-acts over commutative Γ semigroups, notions that generalize classical weak multiplication and comultiplication modules. We establish fundamental properties, provide equivalent characterizations using annihilators and residual Γ ideals, and prove a localization theorem. It is shown that every finitely generated weak Γ multiplication SГ-act is cyclic, hence a multiplication act. Over Γ-artinian Γ semigroups, every weak Γ multiplication SГ-act is cyclic. A categorical duality between finitely generated faithful weak Γ comultiplication SГ-acts and finitely generated faithful multiplication SГ-acts is established for semihereditary Γ semigroups. Several examples delimit the concepts introduced.

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