Minimally complete associative Artin rings

T. V. Pavlova

Sibirskie èlektronnye matematičeskie izvestiâ, Tome 14 (2017), pp. 1238-1247.

Résumé

We study complete associative rings. We give an exhaustive description of minimally complete associative Artin rings.

Détail

Keywords: associative ring, Artin ring, complete ring, reduced ring, minimally complete ring, Galois ring.

Bibliographie

[1] L. M. Martynov, “On concepts of completeness, reducibility, primarity, and purity for arbitrary algebras”, Universal algebra and its applications (Volgograd, September 6–11, 1999), Peremena, Volgograd, 2000, 179–190 (in Russian)

[2] L. M. Martynov, “Completeness, reducibility, primarity and purity for algebras: results and problems”, Siberian Electr. Math. Reports, 13 (2016), 181–241 (in Russian) | MR | Zbl

[3] A. I. Kornev, “On complete modules”, Abelian Groups and Modules, 15, TGU, Tomsk, 2000, 30–37 (in Russian)

[4] V. V. Ovchinnikov, “On minimally complete modules over commutative local rings”, Matematika i Informatika: Nauka i Obrazovanie, 2, OmGPU, Omsk, 2002, 54–56 (in Russian)

[5] T. Ju. Fink, “Embeddability and minimal completeness of finite semigroups”, Matematika i Informatika: Nauka i Obrazovanie, 1, OmGPU, Omsk, 2001, 20–25 (in Russian)

[6] O. V. Knyazev, T. Ju. Fink, “Minimal complete periodic semigroups with zero which the set of nilelements is not subsemigroup”, Matematika i Informatika: Nauka i Obrazovanie, 8, OmGPU, Omsk, 2009, 12–15 (in Russian)

[7] O. V. Knyazev, “On minimally complete commutative epigroups”, Matematika i Informatika: Nauka i Obrazovanie, 10, OmGPU, Omsk, 2011, 6–8 (in Russian)

[8] L. M. Martynov, T. V. Pavlova, “On minimally complete associative rings”, Herald of Omsk University, 1 (2016), 6–13 (in Russian)

[9] T. A. Martynova, “Minimally complete unars”, Matematika i Informatika: Nauka i Obrazovanie, 9, OmGPU, Omsk, 2010, 36–39 (in Russian)

[10] I. N. Herstein, Noncommutative rings, Mir, M., 1972 (in Russian) | MR | Zbl

[11] R. Raghavendran, “Finite associative rings”, Compos. Math., 21 (1969), 195–229 | MR | Zbl

[12] R. S. Wilson, “On the structure of finite rings”, Compozitio Math., 26 (1973), 79–93 | MR | Zbl

[13] R. S. Wilson, “On structure of finite rings. II”, Pasific J. Math., 51 (1974), 317–325 | DOI | MR | Zbl

[14] B. R. McDonald, Finite rings with identity, Pure Appl. Math., 28, Marcel Dekker, New York, 1974 | MR | Zbl

[15] G. Bini, F. Flamini, Finite commutative rings and their applications, Kluwer Academic Publishers, Boston, MA, 2002 | MR | Zbl

[16] V. L. Kurakin, A. S. Kuzmin, A. V. Mikhalev, A. A. Nechaev, “Linear recurring sequences over rings and modules”, Algebra. 2, J. Math. Sci., 76:2 (1995), 2793–2915 | DOI | MR | Zbl

[17] A. A. Nechaev, “Finite rings with applications”, Handb. Algebr., v. 5, Elsevier/North-Holland, Amsterdam, 2008, 213–320 | DOI | MR | Zbl

[18] L. Fuchs, Infinite Abelian groups, v. 2, Mir, M., 1977 (in Russian) | MR

[19] T. V. Pavlova, “Complete associative Artin rings”, Herald of Omsk University, 1 (2005), 17–19 (in Russian)

[20] Siberian Math. J., 48 (2007), 857–862 | DOI | MR | Zbl

[21] T. V. Pavlova, “On reduced associative Artin rings”, Problemy i perspektivy fiz-mat. i tekhn. obrazovaniya, Filial TyumGU v Ishime, Ishim, 2014, 40–46 (in Russian)

[22] T. V. Pavlova, “Complete radical of full ring of matrices over an arbitrary ring”, Matematika i Informatika: Nauka i Obrazovanie, 10, OmGPU, Omsk, 2011, 16–19 (in Russian)

[23] Siberian Math. J., 45 (2004), 504–512 | DOI | MR | Zbl