The Source of $\Gamma$-semiprimeness on $\Gamma$-semigroups

Year 2025, Volume: 17 Issue: 1, 10 - 16, 30.06.2025

Didem Yeşil , Rasie Mekera , Didem Karalarlıoğlu Camcı

https://doi.org/10.47000/tjmcs.1378193

Abstract

Let $S$ be a $\Gamma$-semigroup with zero. We define the $S_{S}^{\Gamma}$ subset of $S$ as $S_{S}^{\Gamma}=\{a\in S \mid a\Gamma (S\Gamma a)=(0)\}.$ This set is called the source of $\Gamma$-semiprimeness of $S$. In this study, we examined some properties of $S_{S}^{\Gamma}$ set and defined $\lvert S_{S}^{\Gamma}\rvert$-idempotent , $\lvert S_{S}^{\Gamma}\rvert$-regular and $\lvert S_{S}^{\Gamma}\rvert$-reduced $\Gamma$-semigroups. We then obtained some results for these newly defined semigroups.

Keywords

Source of $\Gamma$-semiprimeness , $\lvert S_{S}^{\Gamma}\rvert$-idempotent $\Gamma$-semigroup , $\lvert S_{S}^{\Gamma}\rvert$-regular $\Gamma$-semigroup , $\lvert S_{S}^{\Gamma}\rvert$-reduced $\Gamma$-semigroup

References

• Albayrak, B., Yeşil, D., Karalarlıoğlu Camcı D., The source of semiprimeness of semigroups, Journal of Mathematics, 2021(2021), 1–8.
• Jyothi, V., Sarala, Y., Madhusudhana Rao, D., 2Primal Γ-semigroups, IJPT, 9(2017), 30540–30552.
• Saed, I.A., On prime and semiprime Gamma rings with symmetric Gamma n-centralizers, Ibn Al-Haitham International Conference for Pure and Applied Sciences (IHICPS) 9-10 December 2020, Journal of Physics: Conference Series, Baghdad, Iraq, 1879(2021).
• Savithri, S., Gangadhara Rao, A., Achala, L., Pradeep, J.M., Γ-Semigroups in which primary Γ-ideals are prime and maximal, International Journal of Scientific and Innovative Mathematical Research, 5(2017), 36–43.
• Sen, M. K., Saha, N.K., On Γ-semigroup I, Bulletin of the Calcutta Mathematical Society, 78 (1986), 180–186.
• Siripitukted, M., Iampan, A., On the ideal extensions in Γ-semigroups, Kyungpook Mathematical Journal, 48(2008), 585–591.