A Note on Radicals of Paragraded Rings
DOI:
https://doi.org/10.5644/SJM.12.3.04Keywords:
Paragraded ring, prime/Baer radical, Jacobson radicalAbstract
In this paper we prove that there exist paragraded rings which are not graded and we discuss prime and Jacobson radicals of paragraded rings. In particular, we prove that paragraded counterparts of prime and Jacobson radicals are the largest paragraded ideals contained in them.
* This paper was presented at the International Scientific Conference Graded structures in algebra and their applications, dedicated to the memory of Prof. Marc Krasner, IUCDubrovnik, Croatia, September, 22-24, 2016.
Downloads
References
M. Chadeyras, Essai d’une th´eorie noetherienne pour les anneaux commutatifs, dont la graduation est aussi g´en´erale que possible, Bull. de la S.M.F., Suppl´ement, M´emoire No. 22, Paris 1970.
M. Cohen and S. Montgomery, Group-graded rings, smash products, and group actions, Trans. Amer. Math. Soc., 282 (1) (1984), 237–258.
B. J. Gardner and R. Wiegandt, Radical Theory of Rings, Pure and Applied Mathematics 261, Marcel Dekker 2004.
E. Halberstadt, Le radical d’un anneide r´egulier, C. R. Acad. Sci., Paris, S´er. A, Paris, 270 (1970), 361–363.
E. Halberstadt, Th´eorie artinienne homog`ene des anneaux gradu´es `a grades non commutatifs r´eguliers, PhD Thesis, University Piere and Marie Curie, Paris, 1971.
E. Ili´c-Georgijevi´c and M. Vukovi´c, Sheaves of paragraded rings, Sarajevo J. Math., 7 (20) (2011), 153–161.
E. Ili´c-Georgijevi´c and M. Vukovi´c, The Wedderburn–Artin theorem for paragraded rings, Fundam. Prikl. Mat., 19 (6) (2014), 125–139.
N. Jacobson, Structure of rings, Amer. Math. Soc. Coll. Publ., 37, Providence, 1956.
A. V. Kelarev, On groupoid graded rings, J. Algebra 178 (1995), 391–399.
A. V. Kelarev, Ring constructions and applications, Series in Algebra, Vol. 9, World Scientific, 2002.
A. V. Kelarev and A. Plant, Bergman’s lemma for graded rings, Comm. Algebra, 23 (12) (1995), 4613–4624.
M. Krasner, Une g´en´eralisation de la notion de corps-corpo´ıde. Un corpo´ıde remarquable de la th´eorie des corps valu´es, C. R. Acad. Sci., Paris, 219 (1944), 345–347.
M. Krasner, Anneaux gradu´es g´en´eraux, Colloque d’Alg´ebre Rennes, 1980, 209–308.
M. Krasner and M. Vukovi´c, Structures paragradu´ees (groupes, anneaux, modules) I, Proc. Japan Acad., Ser, A, 62 (9) (1986), 350–352.
M. Krasner and M. Vukovi´c, Structures paragradu´ees (groupes, anneaux, modules) II, Proc. Japan Acad., Ser, A, 62 (10) (1986), 389–391.
M. Krasner and M. Vukovi´c, Structures paragradu´ees (groupes, anneaux, modules) III, Proc. Japan Acad., Ser, A, 63 (1) (1987), 10–12.
M. Krasner and M. Vukovi´c, Structures paragradu´ees (groupes, anneaux, modules), Queen’s Papers in Pure and Applied Mathematics, No. 77, Queen’s University, Kingston, Ontario, Canada 1987.
C. N˘ast˘asescu and F. Van Oystaeyen, Methods of Graded Rings, Lecture Notes in Mathematics 1836, Springer, 2004.
M. Vukovi´c, Structures gradu´ees et paragradu´ees, Prepublication de l’Institut Fourier,
Universit´e de Grenoble I, 536 (2001), 1-40.
M. Vukovi´c and E. Ili´c-Georgijevi´c, Paragraded rings and their ideals, Fundam. Prikl.
Mat., 17 (4) (2012), 83–93, J. Math. Sci., New York 191, No. 5 (2013), 654–660.
