NONE 20260316103010615 wseas/wseas content-registration-form+ja@crossref.org content-registration-form WSEAS TRANSACTIONS ON MATHEMATICS 1109-2769 2224-2880 Samer Adnan Jubair Department of Mathematics, College of Education, Al-Qasim Green University, IRAQ Mahdi Sadiq Abaas Department of Mathematics, College of Education, Al-Zahraa University for Women, IRAQ This article introduces and studies Γ-multiplicatively closed subsets R of a Γ-monoid S, developing their foundational properties through definitions and characterization theorems and essential structural results. The work extends classical localization and ideal theory to Γ-semigroups through three principal advances: the localization of 𝑆Г -acts via a congruence relation, the derivation of universal factorization properties for morphisms and the Local-Global Principle for 𝑆Г -act structures. The concept of 𝑅Г -multiplication 𝑆Г -acts is then introduced as a generalization of multiplication 𝑆Г -acts with a complete characterization of their basic properties. The relationship between multiplication 𝑆Г -acts and 𝑅Г -multiplication 𝑆Г -The act is examined. In particular, we prove that every multiplication 𝑆Г -act is an 𝑅Г -multiplication 𝑆Г -act, but the converse does not always hold. Using the concept of idempotent 𝑆Г -subacts, we show that every 𝑆Г -subact is an 𝑅Г -multiplication 𝑆Г -acts. Furthermore, additional concepts such as 𝑅Г -cyclic and 𝑅Г -finite 𝑆Г -Acts are introduced and some of their properties are investigated. Finally, the notion of an 𝑅Г -finite 𝑆Г -Subact is used to study the relationship between 𝑅Г -cyclic and 𝑅Г -multiplication 𝑆Г -acts, and several important properties are discussed. 12 31 2025 12 31 2025 777 77 https://creativecommons.org/licenses/by/4.0/deed.en_US 10.37394/23206.2025.24.77 https://wseas.com/journals/mathematics/2025/b565106-2282.pdf M.K.Sen, "Γ-Semigroup-I," In Proceedings of the International Conference on Algebra and its Application, pp.1-10, 1981. https://share.google/3TX5bO8RtcmQmnCA E M.K.Sen, " Orthodox Γ-semigroups-II," Bull. Calcutta Math. Soc., Vols. VOL.79,, no. NO.6, pp. 331-335, 1987. https://doi.org/10.1155/S016117129000076 X A. Anjaneyulu, A.Gangadhara, D.Madhusudhana, "Prime Γ-Radicals In Γ -Semigroup," International Journal of Mathematical and Engineering, vol. 138, no. 3, pp. 1250-1259, 2011. ISSN:0976- 1411 S.Savithri, G. Rao, A. Achala, J.M.Pradeep, "Γ-Semigroup in which Primary Γ-Ideal are Prime and Maximal," International Journal of Scientific and Innovative Mathematical Research, vol. 5, no. 7, pp. 36-43, 2017. ISSN 2347-3142 accessed date 24 Sep 2025 M.S. Abbas, A. Faris, "Gamma acts," International Journal of Advanced Research,, vol. 4, pp. 1592-1601, 2016. doi:10.21474/IJAR01/747 S. G. M.S. Abbas, "On the gamma spectrum of multiplication gamma acts,," Kuwait Journal of Science, Vols. 48, no. 2, pp. 1-11. https://doi.org/10.48129/kjs.v48i2.10147 S. Kadhim, U. S. AL-Hussani, M. J. Mohammed, "On En- Prime Compactly Packed Acts over Monoid" Baghdad Science Journal, Vols. 22, no. 5, pp. 1630- 1633,2025. https://doi.org/10.21123/bsj.2024.11424 S.A.Jubair, "Cancellanion and Weak Cancellation S-acts" Al-Bahir Journal for Engineering and Pure Sciences, Vols. 3, no. 1, pp. 28-33, 2023. doi.org/10.55810/2313-0083.1032 S.M. Saied, H.K. Zghair, "T (A)- Prime gamma subacts," Journal of Statistics and Management Systems, vol.25, no.8, pp. 2019-2023, 2022. https://doi.org/10.1080/09720529.2022.208 3681 A.N. Fadhil, A.S. Iftichar, "Controllability of impulsive fractional nonlinear control system with Mittag-Leffle kernel in Banach space," Int J Nonlinear Anal Appl, vol. 13, no. 1, pp. 275-328, 2022. https://doi.org/10.22075/ijnaa.2022.6080. S.A.Jubair, M. S. Abbas, "The product of Gam ma Subacts of Multiplication," Journal of Physics: , vol. 1804, pp. 1-7, 202 doi 10.1088/1742- 6596/1804/1/012105