GARDNER PROBLEM REVISITED: FURTHER PROPERTIES OF INDAH RADICAL
Abstract
Radical theory arises naturally from the study of non-commutative rings and plays a central role in the structural analysis of ring classes. Among the radical classes that have received considerable attention are the prime radical β and the IndaH radical , whose relationship is closely related to the Gardner conjecture. While several structural properties of β are well established, the corresponding properties of have remained less clear. In this paper, we investigate the IndaH radical using deductive arguments and structural analysis within the framework of radical theory. In particular, we examine whether satisfies corner-hereditariness, corner-strictness, very corner-hereditariness, and the hereditary phantom corner (HPC) property. We show that possesses all four properties, thereby placing it in closer structural alignment with the prime radical β. The results are obtained under standard assumptions on associative rings and radical classes.
Downloads
References
J. W. Gardner and R. Wiegandt, RADICAL THEORY OF RINGS, 1st ed. New York, NY, USA: Routledge, 2004. doi: https://doi.org/10.1201/9780203913352
M. Mazzotta, B. Rybołowicz, and P. Stefanelli, “DEFORMED SOLUTIONS OF THE YANG–BAXTER EQUATION ASSOCIATED TO DUAL WEAK BRACES,” Ann. Mat. Pura Appl., vol. 204, no. 2, pp. 711–731, 2025. doi: https://doi.org/10.1007/s10231-024-01502-7
A. Russo and M. Viscusi, “CENTRAL ENDOMORPHISMS OF GROUPS AND RADICAL RINGS,” Adv. Group Theory Appl., vol. 17, pp. 129–141, 2023. doi: https://doi.org/10.1142/S021949882550152X
K. Alhazmy, F. A. A. Almahdi, Y. El Haddaoui, and N. Mahdou, “ON SOME SPECIFIC RESULTS OF RINGS WITH PRIME NILRADICAL,” J. Algebra Appl., vol. 24, no. 6, 2025. doi: https://doi.org/10.1142/S0219498823501086
L. Zhang and Z. Yan, “EINSTEIN NILRADICALS ARE COMPLETABLE,” J. Algebra Appl., vol. 22, no. 5, 2023. doi: https://doi.org/10.1142/S0219498823501086
F. Kourki and R. Tribak, “RINGS WHICH ARE NOETHERIAN MODULO A NILPOTENT NILRADICAL,” J. Algebra Appl., 2025. doi: https://doi.org/10.1142/S0219498826501872
N. K. Kim, P. P. Nielsen, and M. Ziembowski, “RADICALS IN POLYNOMIAL RINGS SKEWED BY AN ENDOMORPHISM,” J. Algebra, vol. 674, pp. 117–142, 2025. doi: https://doi.org/10.1016/j.jalgebra.2025.02.044
A. Abouhalaka, “ALMOST RADICALIZATION AND ALMOST RADICAL IDEAL,” J. Algebra Appl., 2025. doi: https://doi.org/10.1142/S0219498826501276
M. Kepczyk, “RINGS WHICH ARE SUMS OF PI SUBRINGS,” J. Algebra Appl., vol. 19, no. 8, 2020. doi: https://doi.org/10.1142/S0219498820501571
M. A. Naghipoor, “ON ZARISKI SPACES OF FREE ACTS OVER SEMIGROUPS,” J. Algebra Appl., vol. 24, no. 6, 2025. doi: 10.1142/S0219498825501415.
S. Trappeniers, “ON TWO-SIDED SKEW BRACES,” J. Algebra, vol. 631, pp. 267–286, 2023. doi: https://doi.org/10.1016/j.jalgebra.2023.05.003
H. Chen, H. K. Kim, T. K. Kwak, and Y. Lee, “RINGS AND RADICALS RELATED TO PRIMARINESS,” J. Algebra Appl., 2024. doi: https://doi.org/10.1142/S0219498825502354
H. I. Blau and C. Wroblewski, “ON STANDARD TABLE ALGEBRAS OF ORDER II,” J. Algebra, vol. 619, pp. 419–458, 2023. doi: https://doi.org/10.1016/j.jalgebra.2022.12.009
A. Zimmermann, “ORE LOCALISATION FOR DIFFERENTIAL GRADED RINGS; TOWARDS GOLDIE’S THEOREM FOR DIFFERENTIAL GRADED ALGEBRAS,” J. Algebra, vol. 663, pp. 48–80, 2025. doi: https://doi.org/10.1016/j.jalgebra.2024.08.032.
W. Qi and X. Zhang, “SOME REMARKS ON NONNIL-COHERENT RINGS AND IF RINGS,” J. Algebra Appl., vol. 21, no. 11, 2022. doi: https://doi.org/10.1142/S0219498822502115
Steven, “THE RELATION BETWEEN GENERALIZED NOTIONS OF RADICAL FORMULA OF A MODULE,” Commun. Algebra, vol. 52, no. 3, pp. 1179–1195, 2024. doi: https://doi.org/10.1080/00927872.2023.2258401
M. D. Ferrari, V. H. Jorge-Perez, and L. C. Merighe, “MIXED MULTIPLICITY AND CONVERSE OF REES’ THEOREM FOR MODULES,” J. Algebra, vol. 664, pp. 484–510, 2025. doi: https://doi.org/10.1016/j.jalgebra.2024.10.048
H. P. T.-Viet, “VARIATIONS ON THE THOMPSON THEOREM,” J. Algebra, vol. 668, pp. 46–74, 2025. doi: https://doi.org/10.1016/j.jalgebra.2024.12.029
C. Dietzel, “BRACES OF ORDER,” J. Algebra Appl., vol. 20, no. 8, 2021. doi: https://doi.org/10.1142/S0219498821501401
A. S. Panasenko, “ON RADICALS OF NOVIKOV ALGEBRAS,” Commun. Algebra, vol. 52, no. 1, pp. 140–147, 2024. doi: https://doi.org/10.1080/00927872.2023.2235420
N. Groenewald, “ON RADICAL IDEALS OF NON-COMMUTATIVE RINGS,” J. Algebra Appl., vol. 22, no. 9, 2023. doi: https://doi.org/10.1142/S0219498823501967
C. E. Ryan, “RINGS WHICH ARE BAER OR QUASI-BAER MODULO A RADICAL,” Commun. Algebra, vol. 49, no. 10, pp. 4557–4564, 2021. doi: https://doi.org/10.1080/00927872.2021.1924185
L. Sawalmeh and M. Saleh, “ON 2-ABSORBING IDEALS OF COMMUTATIVE SEMIRINGS,” J. Algebra Appl., vol. 22, no. 3, 2023. doi: https://doi.org/10.1142/S0219498823500639
M. A. Naghipoor, “ON PRIME AND RADICAL SUBACTS OF ACTS OVER SEMIGROUPS,” Commun. Algebra, vol. 51, no. 5, pp. 1779–1799, 2023. doi: https://doi.org/10.1080/00927872.2022.2143516
N. Ahanjideh, K. Aziziheris, and M. Ghasemi, “THE PRIME-POWER HYPOTHESIS ON THE CODEGREES OF IRREDUCIBLE CHARACTERS,” Commun. Algebra, vol. 53, no. 1, pp. 226–232, 2025. doi: https://doi.org/10.1080/00927872.2024.2374431
P.-H. Lee and E. R. Puczyłowski, “ON REDUCED RINGS WITH PRIME FACTORS SIMPLE,” J. Algebra Appl., vol. 22, no. 1, 2023. doi: https://doi.org/10.1142/S0219498823500287
H. Suárez, A. Chacón, and A. Reyes, “ON NI AND NJ SKEW PBW EXTENSIONS,” Commun. Algebra, vol. 50, no. 8, pp. 3261–3275, 2022. doi: https://doi.org/10.1080/00927872.2022.2028799
M. Z. Aditya and I. M.-Alamsyah, “JACOBSON GRAPH OVER,” in J. Phys. Conf. Ser., vol. 1722, no. 1, p. 012027, 2021. doi: https://doi.org/10.1088/1742-6596/1722/1/012027
S. Humaira, P. Astuti, I. M. Alamsyah, and E. T. Baskoro, “JACOBSON GRAPH OF MATRIX RINGS,” Math. Stat., vol. 11, no. 5, pp. 794–801, 2023. doi: https://doi.org/10.13189/ms.2023.110505
S. Humaira, P. Astuti, I. M.-Alamsyah, and A. Erfanian, “THE MATRIX JACOBSON GRAPH OF FIELDS,” in J. Phys. Conf. Ser., vol. 1538, no. 1, p. 012008, 2020. doi: https://doi.org/10.1088/1742-6596/1538/1/012008
G. Ulucak and S. Isikay, “ON THE UNIT-JACOBSON GRAPH,” J. Algebra Appl., 2024. doi: https://doi.org/10.1142/S0219498825503591
A. K. Bhuniya and P. Sarkar, “ON THE JACOBSON RADICAL OF A SEMIRING,” J. Algebra Appl., 2024. doi: https://doi.org/10.1142/S0219498825503694
S. Humaira, P. Astuti, I. M.-Alamsyah, and A. Erfanian, “THE MATRIX JACOBSON GRAPH OF FINITE COMMUTATIVE RINGS,” Electron. J. Graph Theory Appl., vol. 10, no. 1, pp. 181–197, 2022. doi: https://doi.org/10.5614/ejgta.2022.10.1.12
K. Paykan, M. Habibi, and A. Parsian, “A NOTE ON THE JACOBSON RADICAL OF ORE EXTENSIONS,” Commun. Algebra, vol. 52, no. 11, pp. 4671–4677, 2024. doi: https://doi.org/10.1080/00927872.2024.2356249
S. Hegde, “RINGS WITH 2 IN THE JACOBSON RADICAL AND GENERALIZED BLEACHED CONDITIONS,” Commun. Algebra, vol. 52, no. 12, pp. 5119–5124, 2024. doi: https://doi.org/10.1080/00927872.2024.2364895
X. Mary, “E-SOLID RINGS, STRONGLY IC RINGS AND THE JACOBSON RADICAL,” J. Algebra Appl., vol. 23, no. 8, 2024. doi: https://doi.org/10.1142/S0219498825501518
K. Paykan and M. Habibi, “AN ALTERNATIVE PERSPECTIVE ON JACOBSON RADICAL OF SKEW INVERSE LAURENT SERIES RINGS,” J. Algebra Appl., vol. 24, no. 3, 2025. doi: https://doi.org/10.1142/S0219498825500902
E. S. Miranda and W. Cortes, “WHEN ARE ORE EXTENSIONS OVER NIL RINGS BROWN–MCCOY RADICALS?,” Publ. Math. Debr., vol. 98, no. 3–4, pp. 277–287, 2021. doi: https://doi.org/10.5486/PMD.2021.8774
S. T. Glavatsky, A. Yu. Golubkov, and A. V. Mikhalev, “TOPOLOGICAL JACOBSON RADICALS. III,” J. Math. Sci., vol. 283, no. 6, pp. 849–895, 2024. doi: https://doi.org/10.1007/s10958-024-07316-5
P. W. Prasetyo, et al., “OPTIMIZATION OF ROBOTIC MOVEMENT: APPLYING LIE ALGEBRA TO IMPROVE THE PERFORMANCE OF MBOT2,” Int. J. Adv. Sci. Eng. Inf. Technol., vol. 15, no. 2, 2025. doi: https://doi.org/10.18517/ijaseit.15.2.20501
P. Raj and D. Pal, “LIE-ALGEBRAIC CRITERION FOR STABILITY OF SWITCHED DIFFERENTIAL-ALGEBRAIC EQUATIONS,” in Proc. IFAC-PapersOnLine, vol. 53, no. 2, 2020, pp. 2004–2009. doi: https://doi.org/10.1016/j.ifacol.2020.12.2702
P. Raj and D. Pal, “LIE ALGEBRAIC CRITERIA FOR STABILITY OF SWITCHED SYSTEMS OF DIFFERENTIAL ALGEBRAIC EQUATIONS (DAES),” IEEE Control Syst. Lett., vol. 5, no. 4, pp. 1333–1338, Oct. 2021. doi: https://doi.org/10.1109/LCSYS.2020.3036577
P. Raj and D. Pal, “STABILIZABILITY OF SWITCHED DIFFERENTIAL ALGEBRAIC EQUATIONS: A SOLVABILITY-BASED APPROACH,” IEEE Control Syst. Lett., vol. 8, pp. 1457–1462, 2024. doi: https://doi.org/10.1109/LCSYS.2024.3412864
P. Raj and D. Pal, “ON STABILITY OF SWITCHED DIFFERENTIAL ALGEBRAIC EQUATIONS: A DECOMPOSITION-BASED APPROACH,” IEEE Control Syst. Lett., vol. 8, pp. 358–363, 2024. doi: https://doi.org/10.1109/LCSYS.2024.3382612
H. F.-Jackson, “*-RINGS AND THEIR RADICALS,” Quaest. Math., vol. 8, no. 3, pp. 231–239, 1985. doi: https://doi.org/10.1080/16073606.1985.9631914
H. F.-Jackson, S. Wahyuni, and I. E. Wijayanti, “RADICAL RELATED TO SPECIAL ATOMS REVISITED,” Bull. Aust. Math. Soc., vol. 91, no. 2, pp. 202–210, Apr. 2015. doi: https://doi.org/10.1017/S0004972714000811
P. W. Prasetyo, I. E. Wijayanti, and H. F.-Jackson, “SOME SPECIAL CLASSES WHOSE UPPER RADICAL DETERMINED BY COINCIDES WITH THE UPPER RADICAL DETERMINED BY,” in AIP Conf. Proc., vol. 1707, 2016. doi: https://doi.org/10.1063/1.4940814
P. W. Prasetyo, R. Setyaningsih, I. E. Wijayanti, and H. F.-Jackson, “SOME PROPERTIES OF PRIME ESSENTIAL RINGS AND THEIR GENERALIZATION,” in AIP Conf. Proc., vol. 1746, 2016. doi: https://doi.org/10.1063/1.4958543.
P. W. Prasetyo, I. E. Wijayanti, and H. F.-Jackson, “MODULES AND A SPECIAL CLASS OF MODULES DETERMINED BY THE ESSENTIAL CLOSURE OF THE CLASS OF ALL ∗-RINGS,” JP J. Algebra Number Theory Appl., vol. 39, no. 1, 2017. doi: https://doi.org/10.17654/NT039010011
S. Wahyuni, I. E. Wijayanti, and H. F.-Jackson, “A PRIME ESSENTIAL RING THAT GENERATES A SPECIAL ATOM,” Bull. Aust. Math. Soc., vol. 95, no. 2, pp. 214–218, Apr. 2017. doi: https://doi.org/10.1017/S000497271600085X
P. W. Prasetyo, I. E. Wijayanti, H. F.-Jackson, and J. Repka, “WEAKLY SPECIAL CLASSES OF MODULES,” Math. Stat., vol. 8, no. 2, pp. 23–27, Mar. 2020. doi: https://doi.org/10.13189/ms.2020.081304
H. F.-Jackson, “ON LEFT (RIGHT) STRONG AND LEFT (RIGHT) HEREDITARY RADICALS,” Quaest. Math., vol. 29, no. 3, pp. 329–334, 2006. doi: https://doi.org/10.2989/16073600609486167
E. P. Cojuhari and B. J. Gardner, “RADICALS AND IDEMPOTENTS I,” Commun. Algebra, vol. 49, no. 1, pp. 73–84, 2021. doi: https://doi.org/10.1080/00927872.2020.1793161
P. W. Prasetyo, “NECESSARY AND SUFFICIENT CONDITIONS FOR TO COINCIDE WITH THE PRIME RADICAL,” Ph.D. dissertation, Dept. Math., Universitas Gadjah Mada, Yogyakarta, Indonesia, 2018.
E. P. Cojuhari and B. J. Gardner, “RADICALS AND IDEMPOTENTS II,” Commun. Algebra, vol. 50, no. 11, pp. 4791–4804, 2022. doi: https://doi.org/10.1080/00927872.2022.2075004
Copyright (c) 2026 Puguh Wahyu Prasetyo, Muhammad Ardiyansyah
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.
Authors who publish with this Journal agree to the following terms:
- Author retain copyright and grant the journal right of first publication with the work simultaneously licensed under a creative commons attribution license that allow others to share the work within an acknowledgement of the work’s authorship and initial publication of this journal.
- Authors are able to enter into separate, additional contractual arrangement for the non-exclusive distribution of the journal’s published version of the work (e.g. acknowledgement of its initial publication in this journal).
- Authors are permitted and encouraged to post their work online (e.g. in institutional repositories or on their websites) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published works.
1.gif)
