Structural Characterizations of Primeness in Semigroups

Year 2025, Volume: 6 Issue: 2, 113 - 121, 31.12.2025

Didem Yeşil , Rasie Mekera

https://doi.org/10.54559/amesia.1759955

Abstract

This study investigates the set $P_{S} = \{ b \in S \mid SSb = (0) \}$ for a semigroup $S$ with zero, where $P_{S}$ is referred to as the \emph{source of primeness} of $S$. The paper first explores the fundamental algebraic properties of $P_{S}$, establishing its structural role within the semigroup. It then provides a characterization of $\lvert P_{S}\rvert$-regular, $\lvert P_{S}\rvert$-idempotent, $\lvert P_{S}\rvert$-reduced, and $\lvert P_{S}\rvert$-nonzero-divisor semigroups. Furthermore, the work considers the relationship between the notions of semiprimeness and primeness in semigroups, highlighting how the source of primeness can serve as a tool to analyze and distinguish these concepts. An illustrative example is presented to demonstrate the applicability of the results.

Keywords

Prime semigroup , $\lvert P_{S}\rvert$-regular semigroup , $\lvert P_{S}\rvert$-reduced semigroup , $\lvert P_{S}\rvert$-idempotent semigroup , $\lvert P_{S}\rvert$-nonzero divisor semigroup

References

• M. Bajrami, R. Ramani-Halili, Completely prime ideals of semigroups, Journal of Natural Sciences and Mathematics of UT 5 (9-10) (2020).
• M. Ebrahimpour, R. Nekooei, On generalizations of prime ideals, Communications in Algebra 40 (4) (2012) 1268-1279.
• M. Shabir, N. Kanwal, Prime bi-ideals of semigroups, Southeast Asian Bulletin of Mathematics 31 (4) (2007) 757-764.
• K. P. Shum, Some remarks on topologically semiprime ideals, Czechoslovak Mathematical Journal 34 (1) (1984) 126-129.
• N. Aydın, Ç. Demir, D. Karalarlıoğlu Camcı, The Source of semiprimeness of rings, Communications of the Korean Mathematical Society 33 (4) (2018) 1083-1096.
• B. Albayrak, D. Yeşil, D. Karalarlıoğlu Camcı, The source of semiprimeness of semigroups, Journal of Mathematics 2021 (1) (2021) 1-8.
• R. Mekera, D. Yeşil, A Source of semiprimeness on inverse and completely regular semigroups, Ukrainian Mathematical Journal 76 (8) (2025) 1417-1423.
• D. Yeşil, D. Karalarlıoğlu Camcı, The Source of primeness of rings, Journal of New Theory (41) (2022) 100-104.
• P. A. Grillet, Commutative semigroups, Vol. 2, Springer, 2013.
• P. A. Grillet, Semigroups: An introduction to the structure theory, Routledge, 2017.
• J. M. Howie, Fundamentals of semigroups theory, Vol. 12, Oxford University Press, 1995.
• D. D. Anderson, S. Chun, J. R. Juett, Weakly prime ideals in commutative semigroups, Bulletin of the Korean Mathematical Society 56 (4) (2019) 829-839.
• A. Mahboob, N. M. Khan, Pure $\Gamma$-ideals in $\Gamma$-semigroups, Afrika Matematika 32 (7-8) (2021) 1201-1210.
• H. Wu, Q. Li, On the spectra of commutative semigroups, Semigroup Forum 101 (2) (2020) 465-485.