Geodesic
Radicals of paragraded rings
Fundamentalʹnaâ i prikladnaâ matematika, Tome 24 (2022) no. 2, pp. 3-22.

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This paper is concerned with the theory of paragraded rings, which begins with a series of Krasner and Vuković's notes in Proceedings of the Japan Academy, which first appeared in late 1980s. We present prime and Jacobson radicals, discuss the general Kurosh–Amitsur theory of radicals of paragraded rings, establish that the theorem of Anderson, Divinsky, and Suliński holds for paragraded rings, characterise paragraded normal radicals, and prove that all special paragraded radicals of paragraded rings can be described by appropriate classes of their graded modules. 
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M. Vuković. Radicals of paragraded rings. Fundamentalʹnaâ i prikladnaâ matematika, Tome 24 (2022) no. 2, pp. 3-22. http://geodesic.mathdoc.fr/item/FPM_2022_24_2_a0/

[1] Andrunakievich V. A., Ryabukhin Yu. M., “Spetsialnye moduli i spetsialnye radikaly”, DAN SSSR, 147:6 (1962), 1274–1277 | Zbl

[2] Vukovich M., Ilich-Georgievich E., “Paragraduirovannye koltsa i ikh idealy”, Fundament. prikl. matem., 17:4 (2012), 83–93

[3] Ilich-Georgievich E., Vukovich M., “Teorema Vedderberna–Artina dlya paragraduirovannykh kolets”, Fundament. i prikl. matem., 19:6 (2014), 125–139

[4] Kurosh A. G., “Radikaly kolets i algebr”, Matem. sb., 33:1 (1953), 3–26

[5] Amitsur S. A., “A general theory of radicals. II”, Amer. J. Math., 76, Radicals in Rings and Bicategories (1954), 100–125 | DOI | Zbl

[6] Anderson R., Divinsky N., Suliński A., “Hereditary radicals in associative and alternative rings”, Can. J. Math., 7 (1965), 594–603 | DOI

[7] Balaba I. N., “Special radicals of graded rings”, Bul. Acad. Şti. Rep. Mold. Mat., 44:1 (2004), 26–33 | MR | Zbl

[8] Bourbaki N., Algèbre, Herman, Paris, 1962 | Zbl

[9] Brown B., McCoy N. H., “Radicals and subdirect sum”, Amer. J. Math. Soc., 67 (1947), 46–48 | DOI

[10] Chadeyras M., “Essai d'une théorie noetherienne pour les anneaux commutatifs, dont la graduation est aussi générale que possible”, Bull. Soc. Math. Fr. Suppl. Mém., 22 (1970), 3–143 | MR

[11] Cohen M., Montgomery S., “Group-graded rings, smash products, and group actions”, Trans. Amer. Math. Soc., 282:1 (1984), 237–258 | DOI | MR | Zbl

[12] Fang H., Stewart P., “Radical theory for graded rings”, J. Austral. Math. Soc. Ser. A, 52 (1992), 143–153 | DOI | Zbl

[13] Gardner B. J., Wiegandt R., Radical Theory of Rings, Marcel Dekker, 2004 | Zbl

[14] Halberstadt E., Théorie artinienne homogène des anneaux gradués à grades non commutatifs réguliers, PhD Thesis, Univ. Piere and Marie Curie, Paris, 1971

[15] Ilić-Georgijević E., Vuković M., “Sheaves of paragraded rings”, Sarajevo J. Math., 7(20) (2011), 153–161 | Zbl

[16] Ilić-Georgijević E., Vuković M., “A note on radicals of paragraded rings”, Sarajevo J. Math., 12(25):2 (2016), 307–316 | Zbl

[17] Ilić-Georgijević E., Vuković M., “A note on general radicals of paragraded rings”, Sarajevo J. Math., 12(25):2 (2016), 317–324 | MR | Zbl

[18] Jacobson N., “The radical and semi-simplicity for arbitrary rings”, Amer. J. Math., 78 (1945), 300–320 | DOI

[19] Jespers E., Krempa J., Puczyłowski E. R., “On radicals of graded rings”, Commun. Algebra, 10:17 (1980), 1849–1854 | DOI

[20] Krasner M., “On radicals of graded ringsUne généralisation de la notion de corps-corpoíde. Un corpoíde remarquable de la théorie des corps valués”, C. R. Acad. Sci. Paris, 219 (1944), 345–347 | Zbl

[21] Krasner M., “Anneaux gradués généraux”, Colloq. d'Algèbre (Rennes, 1980), 209–308 | Zbl

[22] Krasner M., Vuković M., “Structures paragraduées (groupes, anneaux, modules). I”, Proc. Japan Acad. Ser. A, 62:9 (1986), 350–352 | DOI | MR | Zbl

[23] Krasner M., Vuković M., “Structures paragraduées (groupes, anneaux, modules). II”, Proc. Japan Acad. Ser. A, 62:10 (1986), 389–391 | DOI | MR | Zbl

[24] Krasner M., Vuković M., “Structures paragraduées (groupes, anneaux, modules). III”, Proc. Japan Acad. Ser. A, 63:1 (1987), 10–12 | DOI | MR | Zbl

[25] Krasner M., Vuković M., Structures paragraduées (groupes, anneaux, modules), Queen's Papers Pure Appl. Math., 77, Queen's University, Kingston, 1987 | MR

[26] Nǎstǎsescu C., van Oystaeyen F., Methods of Graded Rings, Lect. Notes Math., 1836, Springer, 2004

[27] Sands A. D., “On normal radicals”, J. London Math. Soc., 11 (1975), 361–365 | DOI | MR | Zbl

[28] Vuković M., Structures graduées et paragraduées, Prepublication de l'Institut Fourier, No 536, Université de Grenoble I, 2001

[29] Vuković M., “From Krasner's corpoid and Bourbaki's graduations to Krasner's graduations and Krasner–Vuković's paragraduations”, Sarajevo J. Math., 14(27):2 (2018), 175–190 | MR | Zbl