Commutators for near-rings: Huq \({\neq}\) Smith. (English) Zbl 1353.16047
The authors find a counter-example showing that the Huq commutator does not coincide with the Smith commutator for near rings. These commutators coincide for groups and for rings. The Huq commutator has been defined by S. A. Huq in 1968 for semiabelian varieties of universal algebras in a paper published in [Q. J. Math., Oxf. II. Ser. 19, 363–389 (1968; Zbl 0165.03301)] and Smith commutators are defined in the book written by J. D. H. Smith [Mal’cev varieties. Berlin-Heidelberg-New York: Springer-Verlag (1976; Zbl 0344.08002)] in 1976.
Reviewer: Mirela Ştefănescu (Bucureşti)
References:
| [1] | Betsch, G., Kaarli, K.: Supernilpotent radicals and hereditariness of semisimple classes of near-rings. In: Radical Theory (Eger, 1982). Colloq. Math. Soc. J. Bolyai, vol. 38, pp. 47-58. North-Holland, Amsterdam (1985) · Zbl 0583.16029 |
| [2] | Borceux, F., Bourn, D.: Mal’cev, protomodular, homological and semi-abelian categories. Mathematics and its Applications, vol. 566. Kluwer, Dordrecht (2004) · Zbl 1061.18001 |
| [3] | Bourn D.: Normal functors and strong protomodularity. Theory Appl. Categ. 7, 206-218 (2000) · Zbl 0947.18004 |
| [4] | Bourn D., Gran M.: Centrality and normality in protomodular categories. Theory Appl. Categ. 9, 151-165 (2002) · Zbl 1004.18004 |
| [5] | Bourn D., Janelidze G.: Extensions with abelian kernels in protomodular categories. Georgian Math. J. 11, 645-654 (2004) · Zbl 1069.18013 |
| [6] | Bourn, D., Janelidze, G.: Extensions with abelian kernels in protomodular categories. Cah. Topol. Géom. Différ. Catég. L, 211-232 (2009) · Zbl 1187.18011 |
| [7] | Fong Y., Veldsman S., Wiegandt R.: Radical theory in varieties of near-rings in which the constants form an ideal. Comm. Algebra 21, 3369-3384 (1993) · Zbl 0798.16033 · doi:10.1080/00927879308824735 |
| [8] | Fuchs P.: On near-rings in which the constants form an ideal. Bull. Austral. Math. Soc. 39, 171-175 (1989) · Zbl 0655.16026 · doi:10.1017/S0004972700002653 |
| [9] | Gerstenhaber, M.: A categorical setting for the Baer extension theory. In: Proc. Symp. in Pure Math. vol. 17, pp. 50-64. Amer. Math. Soc., Providence (1970) · Zbl 0248.18011 |
| [10] | Gran M., Janelidze G., Ursini A.: Weighted commutators in semi-abelian categories. J. Algebra 397, 643-665 (2014) · Zbl 1305.18040 · doi:10.1016/j.jalgebra.2013.07.037 |
| [11] | Gumm, H. P.: Geometrical methods in congruence modular algebra. Memoirs Amer. Math. Soc. 45, no. 286 (1983) · Zbl 0547.08006 |
| [12] | Hartl M., van der Linden T.: The ternary commutator obstruction for internal crossed modules. Adv. Math. 232, 571-607 (2013) · Zbl 1258.18007 · doi:10.1016/j.aim.2012.09.024 |
| [13] | Higgins P. J.: Groups with multiple operators. Proc. London Math. Soc. 6, 366-416 (1956) · Zbl 0073.01704 · doi:10.1112/plms/s3-6.3.366 |
| [14] | Huq S. A.: Commutator, nilpotency, and solvability in categories. Quart. J. Oxford 19, 363-389 (1968) · Zbl 0165.03301 · doi:10.1093/qmath/19.1.363 |
| [15] | Janelidze G., Kelly G.M.: Central extensions in Mal’tsev varieties. Theory Appl. Categ. 7, 219-226 (2000) · Zbl 0956.08002 |
| [16] | Janelidze G., Márki L., Tholen W.: Semi-abelian categories. J. Pure Appl. Algebra 168, 367-386 (2002) · Zbl 0993.18008 · doi:10.1016/S0022-4049(01)00103-7 |
| [17] | Janelidze G., Pedicchio M.C.: Pseudogroupoids and commutators. Theory Appl. Categ. 8, 408-456 (2001) · Zbl 1008.18006 |
| [18] | Martins-Ferreira N., van der Linden T.: A note on the “Smith is Huq” condition. Appl. Categ. Structures 20, 175-187 (2012) · Zbl 1255.18008 · doi:10.1007/s10485-010-9231-2 |
| [19] | Montoli A.: Action accessability for categories of interest. Theory Appl. Categ. 23, 7-21 (2010) · Zbl 1307.18015 |
| [20] | Smith, J.D.H.: Mal’cev Varieties. Lecture Notes in Mathematics, vol. 554. Springer, Berlin (1976) · Zbl 0344.08002 |
| [21] | Veldsman S.: Supernilpotent radicals of near-rings. Commun. Algebra 15, 2497-2509 (1987) · Zbl 0628.16022 · doi:10.1080/00927878708823549 |
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