Abstract
In this paper, we study atomic elements in the lattices of radical classes. A minimal element of the lattice of all \(N\)-radicals (respectively, supernilpotent radicals and special radicals) is called an \(N\)-atom (respectively, a supernilpotent atom and a special atom). We construct a class of \(N\)-atoms generated by prime essential rings and investigate their structural properties. We show that these \(N\)-atoms are closely related to supernilpotent atoms and special atoms, and we clarify the precise relationships among these three types of atoms. In particular, we prove that for a prime essential ring \(R\), the associated radicals \(\widetilde{\mathit{l}}_{R}\) and \(\overline{\mathit{l}}_{R}\) generate, respectively, a special atom and a supernilpotent atom. This result provides a positive answer to an open question concerning which prime essential rings give rise to atomic elements in the lattices of all $N-$ radical, special radicals and supernilpotent radicals.
Full text article
References
N. Divinsky, J. Krempa, and A. Sulinski, “Strong radical properties of alternative and associative rings,” Journal of Algebra, vol. 17, pp. 369–381, 1971.
B. J. Gardner and R. Wiegandt, Radical Theory of Rings. New York: Marcel Dekker Inc., 2004. https://doi.org/10.1201/9780203913352.
H. France-Jackson, “On atoms of the lattice of supernilpotent radicals,” Quaestiones Mathematicae, vol. 10, pp. 251–255, 1987. https://www.tandfonline.com/doi/abs/10.1080/16073606.1987.9631932.
E. R. Puczylowski and E. Roszkowska, “On atoms of lattices of radicals of associative rings,” Communications in Algebra, vol. 20, pp. 955–977, 1992. https://doi.org/10.1080/00927879208824385.
H. Korolczuk, “A note on the lattice of special radicals,” Bulletin of the International Academy of Polish Sciences, vol. XXIX, pp. 3–4, 1981.
B. J. Gardner and P. N. Stewart, “Prime essential rings,” Proceedings of the Edinburgh Mathematical Society, vol. 34, pp. 241–250, 1991. https://doi.org/10.1017/S0013091500007148.
H. France-Jackson and N. J. Groenewald, “On rings generating supernilpotent and special atoms,” Quaestiones Mathematicae, vol. 28, pp. 471–478, 2005. https://www.tandfonline.com/doi/abs/10.2989/16073600509486141.
S. Wahyuni, I. E. Wijayanti, and H. France-Jackson, “A prime essential ring that generates a special atom,” Bulletin of the Australian Mathematical Society, vol. 95, pp. 214–218, 2016. https://doi.org/10.1017/S000497271600085X.
A. D. Sands, “Radicals and morita context,” Journal of Algebra, vol. 24, pp. 335–345, 1973. https://doi.org/10.1016/0021-8693(73)90143-9.
A. D. Sands, “On normal radicals,” Journal of the London Mathematical Society, vol. 11, pp. 361–365, 1975. https://doi.org/10.1112/jlms/s2-11.3.361.
M. Jaegermann, “Normal radicals,” Fundamenta Mathematicae, vol. 95, pp. 147–155, 1977. https://doi.org/10.4064/fm-95-3-147-155.
M. Jaegerman and A. D. Sands, “On normal radicals, n-radicals and a-radicals,” Journal of Algebra, vol. 50, pp. 337–349, 1978. https://doi.org/10.1016/0021-8693(78)90159-X.
K. I. Beidar and S. Salavova, “On lattices of n-radicals, left strong radicals, left hereditary radicals,” Acta Mathematica Hungarica, vol. 42, pp. 81–95, 1983. (in Russian).
H. France-Jackson, S. Wahyuni, and I. E. Wijayanti, “Radical related to special atoms revisited,” Bulletin of the Australian Mathematical Society, vol. 91, pp. 202–210, 2015. https://doi.org/10.1017/S0004972714000811.
P. W. Prasetyo, I. E. Wijayanti, H. France-Jackson, and J. Repka, “A note on β-classes,” Asian-European Journal of Mathematics, vol. 19, no. 2, p. 2540011, 2026. https://doi.org/10.1142/S179355712540011X.
P. W. Prasetyo, H. Marubayashi, and I. E. Wijayanti, “On the restricted graded Jacobson radical of rings of morita context,” Turkish Journal of Mathematics, vol. 46, no. 5, pp. 1985–1993, 2022. https://doi.org/10.55730/1300-0098.3246.
H. Marubayashi, P. W. Prasetyo, and I. E. Wijayanti, “Rings of morita context which are dubrovin valuation rings,” Journal of Algebra and Its Applications, vol. 22, no. 5, p. 2350229, 2023. https://doi.org/10.1142/S0219498823502298.
P. W. Prasetyo, I. E. Wijayanti, H. France-Jackson, and J. Repka, “Weakly special classes of modules,” Mathematics and Statistics, vol. 8, no. 2A, pp. 23–27, 2020. https://doi.org/10.13189/ms.2020.081304.
D. A. Yuwaningsih, I. E. Wijayanti, and P. W. Prasetyo, “On (r, s)-module homomorphisms,” Journal of Physics: Conference Series, vol. 1188, no. 1, p. 012114, 2019. https://doi.org/10.1088/1742-6596/1188/1/012114.
P. W. Prasetyo, D. A. Yuwaningsih, and B. A. Nurnugroho, “Algebraic structure of supernilpotent radical class constructed from a topology thychonoff space,” Al-Jabar: Jurnal Pendidikan Matematika, vol. 11, no. 2, pp. 331–338, 2020. https://doi.org/10.24042/ajpm.v11i2.6897.
W. D. Burgess and R. Raphael, “Tychonoff spaces and a ring theoretic order on c(x),” Topology and its Applications, vol. 279, p. 107250, 2020. https://doi.org/10.1016/j.topol.2020.107250.
P. W. Prasetyo and M. Ardiyansyah, “Gardner problem revisited: Further properties of indah radical,” BAREKENG: Journal of Mathematics and Its Applications, vol. 20, pp. 2339–2348, Sept. 2026. https://doi.org/10.30598/barekengvol20iss3pp2339-2348.
E. R. Puczylowski, “On normal classes of rings,” Communications in Algebra, vol. 20, pp. 2999–3013, 1992. https://doi.org/10.1080/00927879208824502.
Authors
Copyright (c) 2026 Journal of the Indonesian Mathematical Society
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
Article Details
