Principally 1-Absorbing Right Primary Ideals

Year 2025, Issue: 51, 33 - 41, 30.06.2025

Hatice Çay

https://doi.org/10.53570/jnt.1671463

Abstract

This paper first defines the 1-absorbing version of principally right primary ideals (P1ARP ideals), generalizing prime ideals, for noncommutative rings. It then investigates various properties of this ideal structure in different ring settings. It obtains some essential results in ring extensions, such as homomorphic images, product rings, local rings, and idealization. While this study enables the obtaining of original results due to structural differences between commutative and noncommutative rings, it shows that some properties valid in commutative rings are preserved. Finally, the paper concludes by discussing two open problems that could guide future studies.

Keywords

$1$-Absorbing primary ideals , noncommutative rings , prime ideals , pseudo-radicals

References

• A. Badawi, E. Y. Celikel, On 1-absorbing primary ideals of commutative rings, Journal of Algebra and its Applications 19 (6) (2020) 2050111.
• R. Nikandish, M. J. Nikmehr, A. Yassine, Some results on 1-absorbing primary and weakly 1-absorbing primary ideals of commutative rings, Bulletin Korean Mathematical Society 58 (5) (2021) 1069-1078.
• G. F. Birkenmeier, J. Y. Kim, J. K. Park, Right primary and nilary rings and ideals, Journal of Algebra 378 (2013) 133-152.
• A. Abouhalaka, H. Çay, N. Groenewald, On weakly S-$\rho$-ideals in noncommutative rings, Gulf Journal of Mathematics 18 (2) (2024) 1-15.
• A. Abouhalaka, S. Findik, Extension of almost primary ideals to noncommutative rings and the generalization of nilary ideals, Mathematics 11 (8) (2023) 1917.
• A. Abouhalaka, S. Findik, Generalization of primary ideal associated with a special radical class, Sao Paulo Journal of Mathematical Sciences 18 (1) (2024) 159-178.
• H. Çay, A. Abouhalaka, B. A. Ersoy, Generalization of ρ-ideals associated with an m-system and a special radical class, Ricerche di Matematica (2024) (2024) 1-16.
• O. Almallah, N. Jarboui, H. M. Al-Noghashi, Weakly and completely nilary ideals, JP Journal of Algebra, Number Theory and Applications 40 (5) (2018) 691-700.
• O. Al-Mallah, G. Birkenmeier, H. Alnogashi, Nilary group rings and algebras, Turkish Journal of Mathematics 47 (4) (2023) 1051-1072.
• N. J. Groenewald, On 2-absorbing and weakly 2-absorbing principally right primary ideals, Journal of Algebra and Related Topics 9 (2) (2021) 47-67.
• N. Groenewald, On weakly right primary ideals, Palestine Journal of Mathematics 11 (4) (2022) 282-292.
• N. Groenewald, 1-Absorbing prime ideals and weakly 1-absorbing prime ideals in noncommutative rings, Sao Paulo Journal of Mathematical Sciences 17 (2) (2023) 871-887.
• C. Gorton, H. E. Heatherly, Generalized primary rings and ideals, Mathematica Pannonica 17 (2006) 17-28.
• V. Stefan, A note on the radicals of idealizations, Southeast Asian Bulletin of Mathematics 32 (3) (2008) 545-551.