Completeness, reducibility, primarity and purity for algebras: results and problems

L. M. Martynov

Sibirskie èlektronnye matematičeskie izvestiâ, Tome 13 (2016), pp. 181-241.

Résumé

We give a survey of researches of the author’s concepts (1995, 1996) completeness, reducibility, primarity, purity and problems about them (2000) for universal algebras, groups, modules, monoassociative algebras, semigroups, lattices and unary.

Détail

Keywords: universal algebra; variety of algebras; complete algebra; reduced algebra; periodic algebra; primary algebra; pure subalgebra.

Bibliographie

[1] Russian Mathematical Surveys, 33:2 (1978), 155–193 | DOI | MR | Zbl | Zbl

[2] N. Bass, “Finistic dimension and a homological generalization of semiprimary rings”, Trans. Amer. Math. Soc., 95:3 (1960), 466–488 | DOI | MR | Zbl

[3] L. Bican, “Kulikov's criterian for modules”, J. reine and angew., 288 (1976), 154–159 | MR | Zbl

[4] L. Bican, “The structure of primary modules”, Acta Univ. Carol. Math. et. phys., 17:2 (1977), 3–12 | MR

[5] S. N. Chernikov, “Groups with systems of complemented subgroups”, Mat. Sbornik., 35 (77):1 (1954), 93–128 (in Russian) | MR

[6] T. Ju. Fink, “Finite complete semigroups”, Estestv. Nauki i Èkologiya, 4, OmGPU, Omsk, 1999, 8–14 (in Russian)

[7] T. Ju. Fink, “Finite semigroups with greatest complete subsemigroups”, Matematika i Informatika: Nauka i Obrazovanie, 2, OmGPU, Omsk, 2002, 28–34 (in Russian)

[8] T. Ju. Fink, “Embeddability and minimal completeness of finite semigroups”, Matematika i Informatika: Nauka i Obrazovanie, 1, OmGPU, Omsk, 2002, 20–25 (in Russian)

[9] T. Ju. Fink, “Splitable pseudovarieties of finite semigroups”, Vestn. Omsk. Univers., 2005, no. 4, 33–35 (in Russian)

[10] T. Ju. Fink, “Pseudovarieties of finite semigroups having the complete radical”, Sib. Èlectron. Matem. Izv., 5 (2008), 673–684 (in Russian) | MR | Zbl

[11] T. Ju. Fink, Complete, reduced, and primary finite semigroups, Dissert., Omsk, 2006 (in Russian)

[12] L. Fuchs, A. Kertesz, T. Szele, “Abelian groups in which every serving subgroups is a direct summand”, Publ. Math., 3 (1953), 95–105 | MR

[13] L. A. Skornyakov (ed.), General algebra, v. 1, Nauka, M., 1990 (in Russian) | MR | Zbl

[14] L. A. Skornyakov (ed.), General algebra, v. 2, Nauka, M., 1991 (in Russian) | MR | Zbl

[15] G. Grätzer, General theory of lattices, Translation from English, ed. D. M Smirnov, Mir, M., 1981 (in Russian) | MR

[16] R. Hamsher, “Commutative rings over which every module has a maximal submodule”, Proc. Amer. Math. Soc., 18 (1967), 1133–1137 | DOI | MR | Zbl

[17] N. Jacobson, Structure of rings, Izdat. Inostran. Liter., M., 1961 (in Russian) | MR

[18] Kibern. sb., 2 (1961), 41–52 | DOI | MR | MR | Zbl

[19] K. Katô, “On abelians groups every subgroup in wich is neat subgroup”, Comment. Math. Univ. St. Pauli., 15 (1966/67), 117–118 | MR

[20] O. V. Knyazev, “On pure algebras”, Vestn. Omsk. Univers., 2001, no. 3, 18–20 (in Russian) | Zbl

[21] O. V. Knyazev, “On pure algebras with distinguished idempotent”, Matematika i Informatika: Nauka i Obrazovanie, 1, OmGPU, Omsk, 2002, 17–19 (in Russian)

[22] O. V. Knyazev, “On absolutely pure algebras with distinguished idempotent”, Matematika i Informatika: Nauka i Obrazovanie, 2, OmGPU, Omsk, 2002, 23–25 (in Russian)

[23] O. V. Knyazev, “On pure algebras and retracts”, Actual problems of contemporary Science and Education, Collection of Scientific Works on Materials of the International Scientific-Practical Conference, In 5 Parts (30 April 2015), v. I, «AR-Consalt», M., 2015, 32–34 (in Russian)

[24] O. V. Knyazev, “On pure cyclic subgroups of groups”, Matematika i Informatika: Nauka i Obrazovanie, 5, OmGPU, Omsk, 2006, 24–26 (in Russian)

[25] O. V. Knyazev, “Semigroups with hereditarly pure semigroups”, Izvestiya Ural. Univers., Mathematika i Mekhanika, 2005, no. 8(38), 69–79 (in Russian) | MR | Zbl

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

[27] O. V. Knyazev, “Hereditarily pure monoids”, Sib. Èlectron. Mat. Izv., 2 (2005), 83–87 (in Russian) | MR | Zbl

[28] O. V. Knyazev, “Hereditarily pure commutative monoids”, Matematika i Informatika: Nauka i Obrazovanie, 3, OmGPU, Omsk, 2003, 16–19 (in Russian)

[29] O. V. Knyazev, “On simple under purity monoids”, Èlectron. Nauch. Zhurn. «Vestn. OmGPU», 2007 (in Russian)

[30] O. V. Knyazev, “On hereditarily pure semigroups”, Vestn. Omsk. Univers., 2003, no. 3, 13–15 (in Russian) | MR

[31] O. V. Knyazev, “Semigroups with zero without pure cyclic subsemigroups”, Matematika i Informatika: Nauka i Obrazovanie, 6, OmGPU, Omsk, 2007, 15–17 (in Russian)

[32] O. V. Knyazev, “On simple under purity semigroups with zero”, Matematika i Informatika: Nauka i Obrazovanie, 7, OmGPU, Omsk, 2008, 19–21 (in Russian)

[33] O. V. Knyazev, “On complete nilsemigroups”, Matematika i Informatika: Nauka i Obrazovanie, 8, OmGPU, Omsk, 2009, 10–12

[34] O. V. Knyazev, “On minimally complete commutative nilsemigroups”, Matematika i Informatika: Nauka i Obrazovanie, 9, OmGPU, Omsk, 2010, 12–14 (in Russian)

[35] 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)

[36] O. V. Knyazev, T. Ju. Fink, “Minimal complete finite semigroups with zero which the set of nilelements is subsemigroup”, Matematika i Informatika: Nauka i Obrazovanie, 9, OmGPU, Omsk, 2010, 14–18 (in Russian)

[37] O. V. Knyazev, T. Ju. Fink, “On simple under purity periodic semigroups with zero”, International Conference MAL'TSEV MEETING dedicated to the 100th anniversary of Anatoli\u{i} Ivanovich Mal'tsev, Collection of Abstracts (August 24–28, 2009), Novosibirsk, 2009, 58 (In Russian)

[38] O. V. Knyazev, Clifford semigroups with pure subsemigroups, Rukopis' predstavlena Sib. Mat. Zh. Dep. VINITI 30 marta 2000 g., No 861-V00 (in Russian)

[39] O. V. Knyazev, “Simple under purity completelly regular semigroups”, Vestn. Omsk. Univers., 2006, no. 4, 9–11 (in Russian)

[40] Yu. A. Kolmakov, “Group varieties whose members satisfy some property close to solvability”, Mat. Zametki, 35:5 (1984), 735–738 (in Russian) | MR | Zbl

[41] L. A. Koifman, “Rings over which each module has a maximal submodule”, Mat. Zametki, 7:3 (1970), 359–367 (in Russian) | MR | Zbl

[42] A. I. Kornev, “On modules with pure submodules”, Universal algebra and its applications, Works of the international seminar (Volgograd, September 6–11, 1999), Peremena, Volgograd, 2000, 144–152 (in Russian)

[43] A. I. Kornev, “Purely simple modules of reduced varieties of modules over commutative rings”, Vestn. Omsk. Univers., 2000, no. 4, 13–15 (in Russian) | MR | Zbl

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

[45] A. I. Kornev, “Rings over which all modules are absolutely pure”, Matematika i Informatika: Nauka i Obrazovanie, 1, OmGPU, Omsk, 2002, 29–31 (in Russian)

[46] A. I. Kornev, “Pure injective modules”, Matematika i Informatika: Nauka i Obrazovanie, 2, OmGPU, Omsk, 2002, 26–28 (in Russian)

[47] A. I. Kornev, Absolutely pure and purely simple modules, Dissert., Omsk, 2001 (in Russian)

[48] A. I. Kornev, T. V. Pavlova, “Finite complete associative rings”, Matematika i Informatika: Nauka i Obrazovanie, 1, OmGPU, Omsk, 2002, 43–45 (in Russian)

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

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

[51] A. G. Kurosh, The theory of groups, Nauka, M., 1967 (in Russian) | MR

[52] A. G. Kurosh, “Radicals in the group theory”, Sib. Mat. Zh., 3 (1962), 912–931 (in Russian) | MR | Zbl

[53] A. G. Kurosh, “Radicals of rings and algebras”, Mat. Sbornik, 33 (1953), 13–26 (in Russian) | MR | Zbl

[54] I. V. L'vov, E. I. Khukhro, “Comments to Question 5.56”, Kourovskaya tetrad' (nereshen. zadachi teorii grupp), Novosibirsk, 1978, 73 (in Russian)

[55] A. I. Mal'cev, “On the multiplication of classes of algebraic systems”, Sib. Mat. Zh., 8 (1967), 346–365 (in Russian) | MR

[56] A. I. Mal'cev, Algebraic systems, Nauka, M., 1970 (in Russian) | MR | Zbl

[57] L. M. Martynov, “Pirmary and reduced varieties of semigrops”, International conference «Semigroups and their appendices, including semigroup rings» in honour E. L. Lyapin, Abstracts of reports (St. Petersburg, June 19–30, 1995), Ekaterinburg, 1995, 38 (In Russian)

[58] L. M. Martynov, “On notions of completeness, solvability, primarity, reducibility and purity for arbitrary algebras”, International conference on Modern Algebra and Its Applications, Schudule and Abstracts (Vanderbilt Univ., Nashville, Tennessee, May 14–18, 1996), 79–80

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

[60] Algebra and Logic, 29:2 (1990), 113–124 | DOI | MR

[61] Soviet Mathematics (Izvestiya VUZ. Matematika), 33:6 (1989), 87–89 | MR

[62] L. M. Martynov, “On relatively complete algebras”, Vestn. Omsk. Univers., 2000, no. 3, 9–11 (in Russian)

[63] L. M. Martynov, “On complete and reduced algebras”, Matematika i Informatika: Nauka i Obrazovanie, 3, OmGPU, Omsk, 2003, 3–8 (in Russian)

[64] L. M. Martynov, “One radical algebras with a property of transversality by minimal varieties”, Vestn. Omsk. Univers., 2004, no. 2, 19–21 (in Russian)

[65] L. M. Martynov, “On primary and reduce varieties of modules”, Vestn. Omsk. Univers., 1999, no. 4, 29–31 (in Russian)

[66] L. M. Martynov, “On attainability and solvability for varietis of modules”, Issledov. algebraic. system, UrGU, Sverdlovsk, 1984, 56–74 (in Russian) | MR

[67] Mathematical Notes, 45:1 (1989), 45–48 | DOI | MR

[68] Siberian Math. J., 42 (2001), 91–98 | DOI | MR | Zbl

[69] L. M. Martynov, “Hereditarily pure associative rings”, Matematika i Informatika: Nauka i Obrazovanie, 9, OmGPU, Omsk, 2010, 25–36 (in Russian)

[70] Algebra Logic, 50 (2012), 526–538 | DOI | MR | Zbl

[71] L. M. Martynov, “On hereditary pure associative algebras over a Dedekind ring”, Sib. Èlectron. Matem. Izv., 10 (2013), 475–490 (in Russian) | MR | Zbl

[72] L. M. Martynov, “Primary varieties of semigroups”, Matematika i Informatika: Nauka i Obrazovanie, 1, OmGPU, Omsk, 2002, 3–9 (in Russian)

[73] Russian Mathematics (Izvestiya VUZ. Matematika), 48:2 (2004), 73–76 | MR

[74] L. M. Martynov, “Varieties in which each semigroup is reduced”, Matematika i informatika: nauka i obrazovanie, 4, OmGPU, Omsk, 2004, 13–21 (in Russian)

[75] L. M. Martynov, “Varieties of commutative semigroups having the complete radical”, Vestn. Omsk. Univers., 2014, no. 4, 14–18 (in Russian)

[76] L. M. Martynov, T. Ju. Fink, “On primary semigrops”, Vestn. Omsk. Univers., 2002, no. 3, 18–20 (in Russian) | Zbl

[77] L. M. Martynov, T. Ju. Fink, “Splittable varieties of groups and semigroups”, Vestn. Omsk. Univers., 2013, no. 2, 32–36 (in Russian)

[78] L. M. Martynov, O. V. Knyazev, “On finite groups in which every pure subgroup is a direct factor”, Vestn. Omsk. Univers., 2014, no. 2, 25–26 | MR

[79] L. M. Martynov, T. V. Pavlova, “On minimally complete associative rings”, International Conference MAL'TSEV MEETING dedicated to 75th anniversary of Yuri\u{i} L. Ershov, Collection of Abstracts (May 3–7, 2015), NGU, Novosibirsk, 166 (In Russian)

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

[81] T. A. Martynova, “Complete unars”, Matematika i Informatika: Nauka i Obrazovanie, 10, OmGPU, Omsk, 2011, 11–16 (in Russian)

[82] A. P. Mishina, L. A. Skornyakov, Abelian groups and moduls, Nauka, M., 1969 (in Russian) | MR

[83] V. N. Neumann, H. Neumann, P. M. Neumann, “Wreath products and varieties of groups”, Math. Z., 80:1 (1962), 44–62 | DOI | MR | Zbl

[84] V. V. Ovchinnikov, “On rings over which each module is reduced”, Abelian Groups and Modules, 15, TGU, Tomsk, 2000, 46–54 (in Russian)

[85] 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)

[86] T. V. Pavlova, “Complete associative Artin rings”, Vestn. Omsk. Univers., 2005, no. 1, 17–19 (in Russian)

[87] T. V. Pavlova, “On reduced associative Artinian rings”, Problems and Prospects of Phys.-Math. and Techn. Education, Filial TyumGU in Ishim, Ishim, 2014, 40–46 (in Russian)

[88] 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)

[89] F. Pastijn, M. Volkov, “Minimal Nonnryptin e-Varieties of regular semigroups”, J. Algebra, 184:3 (1996), 881–896 | DOI | MR | Zbl

[90] Yu. P. Razmyslov, “An example of unsolvable just non-cross varieties of groups”, Algebra i Logika, 11:2 (1972), 186–205 (in Russian) | DOI | MR | Zbl

[91] V. N. Rudakov, “On atomically complete lattices”, Matematika i Informatika: Nauka i Obrazovanie, 10, OmGPU, Omsk, 2011, 19–22 (in Russian)

[92] V. N. Rudakov, “Reduced varieties of unary”, Vestn. Omsk. Univers., 2013, no. 2, 45–47 (in Russian)

[93] M. V. Sapir, E. V. Suhanov, “On varieties of periodic semigroups”, Izv. Vyssh. Uchebn. Zaved., Matem., 4 (1981), 48–55 (in Russian) | MR | Zbl

[94] Siberian Mathematical Journal, 12 (1971), 986–998 | DOI | MR | Zbl

[95] L. N. Ševrin, L. M. Martynov, “Attainability and solvability for classes of algebras”, Semigroups, Coll. Math. Soc. J. Bolyai, 39, eds. G. Poliak, St. Schwarz, O. Steinfeld, Amsterdam–Oxford–New York, 1985, 397–459 | MR

[96] A. L. Shmel'kin, “The semigroup of group varieties”, Dokl. Akadem. Nauk SSSR, 149:3 (1963), 543–545 (in Russian) | MR | Zbl

[97] D. M. Smirnov, Varieties of algebras, VO «Nauka», Novosibirsk, 1992 (in Russian) | MR | Zbl

[98] S. N. Subramanian, T. R. Sundararaman, “Precomplete varieties of semigroups”, Semigroup Forum, 8:3 (1974), 278–281 | DOI | MR | Zbl

[99] T. R. Sundararaman, “Precomplete varieties of R-algebras”, Alg. Univers., 5:1 (1975), 397–405 | DOI | MR | Zbl

[100] A. Tarski, “Equationally complete rings and relation algebras”, Indag. Math., 18 (1956), 39–46 | DOI | MR

[101] A. A. Tuganbaev, “Maximal submodules and locally perfect rings”, Matem. Zametki, 64:1 (1998), 136–142 (in Russian) | DOI | MR | Zbl

[102] A. A. Tuganbaev, “Primitively pure submodules and primitively divisible modules”, J. Math. Sci., 110:3 (2002), 2746–2754 | DOI | MR | Zbl

[103] M. R. Vaughan-Lee, J. Wiegold, “Countable locally nilpotent groups of finite exponent without maximal subgroups”, Bull. Math. Soc., 13:1 (1981), 45–46 | MR | Zbl

[104] M. V. Volkov, “Separation of the radical in ring varieties”, Acta Sci. Math., 46 (1983), 73–75 | MR | Zbl

[105] K. A. Zhevlakov, A. M. Slin'ko, I. P. Shestakov, A. I. Shirsnov, Rings close to associative rings, Nauka, M., 1978 (in Russian)

[106] I. V. Znaeva, “Hereditarly pure Lie algebras”, Matematika i Informatika: Nauka i Obrazovanie, 6, OmGPU, Omsk, 2007, 12–15 (in Russian)

[107] I. V. Znaeva, “On the complete radical matrix Lie algebras”, Matematika i Informatika: Nauka i Obrazovanie, 7, OmGPU, Omsk, 2008, 13–18 (in Russian)

[108] O. Zariski, P. Samuel, Commutative algebra, v. 1, Izdat. Inostran. Liter., M., 1963 (in Russian)