Structural Characterizations of Primeness in Semigroups
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
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.
Details
Primary Language
English
Subjects
Algebra and Number Theory
Journal Section
Research Article
Publication Date
December 31, 2025
Submission Date
August 6, 2025
Acceptance Date
September 29, 2025
Published in Issue
Year 2025 Volume: 6 Number: 2
APA
Yeşil, D., & Mekera, R. (2025). Structural Characterizations of Primeness in Semigroups. Amesia, 6(2), 113-121. https://doi.org/10.54559/amesia.1759955
AMA
1.Yeşil D, Mekera R. Structural Characterizations of Primeness in Semigroups. Amesia. 2025;6(2):113-121. doi:10.54559/amesia.1759955
Chicago
Yeşil, Didem, and Rasie Mekera. 2025. “Structural Characterizations of Primeness in Semigroups”. Amesia 6 (2): 113-21. https://doi.org/10.54559/amesia.1759955.
EndNote
Yeşil D, Mekera R (December 1, 2025) Structural Characterizations of Primeness in Semigroups. Amesia 6 2 113–121.
IEEE
[1]D. Yeşil and R. Mekera, “Structural Characterizations of Primeness in Semigroups”, Amesia, vol. 6, no. 2, pp. 113–121, Dec. 2025, doi: 10.54559/amesia.1759955.
ISNAD
Yeşil, Didem - Mekera, Rasie. “Structural Characterizations of Primeness in Semigroups”. Amesia 6/2 (December 1, 2025): 113-121. https://doi.org/10.54559/amesia.1759955.
JAMA
1.Yeşil D, Mekera R. Structural Characterizations of Primeness in Semigroups. Amesia. 2025;6:113–121.
MLA
Yeşil, Didem, and Rasie Mekera. “Structural Characterizations of Primeness in Semigroups”. Amesia, vol. 6, no. 2, Dec. 2025, pp. 113-21, doi:10.54559/amesia.1759955.
Vancouver
1.Didem Yeşil, Rasie Mekera. Structural Characterizations of Primeness in Semigroups. Amesia. 2025 Dec. 1;6(2):113-21. doi:10.54559/amesia.1759955