The Source of Primeness of Rings

Abstract

In this study, we define a new concept, i.e., source of primeness of a ring $R$, as $P_{R} := \bigcap_{a\in R} S_{R}^{a}$ such that $S_{R}^{a}:=\{b\in R \mid aRb=(0)\}$. We then examine some basic properties of $P_{R}$ related to the ring’s idempotent elements, nilpotent elements, zero divisor elements, and identity elements. Finally, we discuss the need for further research.

Keywords

• Prime ring
• semiprime ring
• source of primeness
• source of semiprimeness

References

1. N. Aydın, Ç. Demir, D. Karalarlıoğlu Camcı, The Source of Semiprimeness of Rings, Communications of the Korean Mathematical Society 33 (2018) 1083-1096.
2. B. Albayrak, D. Yeşil, D. Karalarlıoğlu Camcı, The Source of Semiprimeness of Semigroups, Journal of Mathematics, 2021 (2021), Article ID 4659756, 8 pages.
3. N. H. McCoy, The Theory of Rings, The Macmillan Company New York, Collier-Macmillan Ltd, London, 1964.
4. T. Y. Lam, A First Course in Noncommutative Rings, Graduate Texts in Mathematics, Volume 131, Springer-Verlag, New York, 1991.
5. T. Y. Lam, Lectures on Modules and Rings, Graduate Texts in Mathematics, 189, Springer-Verlag, New York, 1999.
6. S. Ali, N. A. Dar, On Centralizers of Prime Rings with Involution, Bulletin of the Iranian Mathematical Society 41 (6) (2015) 1465-1475.
7. H. Chimal-Dzul, S. Szabo, Classification of Nonsemicommutative Rings, Acta Mathematica Hungarica 165 (1) (2021) 48-62.
8. A. Tarizadeh, M. Aghajani, On Purely Prime Ideals with Applications, Communications in Algebra 49 (2) (2021) 824-835.