Taiwanese Journal of Mathematics


Gerhard Wendt

Full-text: Open access


We study the structure of $0$-primitive near-rings and are able to answer an open question in the theory of minimal ideals in near-rings to the negative, namely if the heart of a zero symmetric subdirectly irreducible near-ring is subdirectly irreducible again. Also, we will be able to classify when a simple near-ring with an identity and containing a minimal left ideal is a Jacobson radical near-ring. Such near-rings are known to exist but have unusual properties. Along the way we prove results on minimal ideals and left ideals in near-rings which so far were known to hold or have been established in the DCCN case, only.

Article information

Taiwanese J. Math., Volume 19, Number 3 (2015), 875-905.

First available in Project Euclid: 4 July 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 16Y30: Near-rings [See also 12K05]

primitive near-rings left ideals minimal ideals simple near-rings subdirectly irreducible near-rings


Wendt, Gerhard. $0$-PRIMITIVE NEAR-RINGS, MINIMAL IDEALS AND SIMPLE NEAR-RINGS. Taiwanese J. Math. 19 (2015), no. 3, 875--905. doi:10.11650/tjm.19.2015.5077. https://projecteuclid.org/euclid.twjm/1499133667

Export citation


  • G. Birkenmeier and H. Heatherly, Minimal ideals in near-rings, Comm. Algebra, 20(2) (1992), 457-468.
  • J. F. T. Hartney and A. M. Matlala, Structure theorems for the socle-ideal of a near-ring, Comm. Algebra, 36(3) (2008), 1140-1152.
  • H. Heatherly, Localized distributivity conditions, Near-rings and near-fields (Fredericton, NB, 1993), 13-29, Math. Appl., 336, Kluwer Acad. Publ., Dordrecht, 1995.
  • H. Heatherly and G. Mason, Ideals in distributively generated nearrings, Publ. Math. Debrecen, 69(1-2) (2006), 121-135.
  • K. Kaarli, On non-zerosymmetric near-rings with minimum condition, Nearrings,nearfields and K-loops, (Hamburg, 1995), 21-33, Math. Appl., 426, Kluwer Acad. Publ., Dordrecht, 1997.
  • K. Kaarli, Minimal ideals in near-rings (Russian), Tartu Riikl. l. Toimetised Vih., 366 (1975), 105-142.
  • K. Kaarli, On minimal ideals of distributively generated near-rings, Contributions to general algebra, 7 (Vienna, 1990), 201-204, Hölder-Pichler-Tempsky, Vienna, 1991.
  • K. Kaarli, On Jacobson type radicals of near-rings, Acta Math. Hungar., 50(1-2) (1987), 71-78.
  • W. F. Ke and J. H. Meyer, Matrix near-rings and 0-primitivity, Monatsh. Math., 165(3-4) (2012), 353-363.
  • T. Y. Lam, A first course in noncommutative rings, Second edition, Graduate Texts in Mathematics, 131. Springer-Verlag, New York, 2001. xx+385 pp. ISBN: 0-387-95183-0.
  • C. J. Maxson and A. P. J. van der Walt, Centralizer near-rings over free ring modules, J. Austral. Math. Soc. Ser. A, 50(2) (1991), 279-296.
  • P. Fuchs, C. J. Maxson, A. P. J. van der Walt and K. Kaarli, Centralizer near-rings determined by PID-modules II, Period. Math. Hungar., 26(2) (1993), 111-114.
  • J. D. P. Meldrum, Near-rings and their links with groups, Research Notes in Mathematics, 134. Pitman (Advanced Publishing Program), Boston, MA, 1985. x+275 pp. ISBN: 0-273-08701-0.
  • G. Pilz, Near-rings, The theory and its applications, Second edition. North-Holland Mathematics Studies, 23. North-Holland Publishing Co., Amsterdam, 1983. xv+470 pp. ISBN: 0-7204-0566-1.
  • S. D. Scott, Minimal ideals of near-rings with minimal condition, J. London Math. Soc., 8(2) (1974), 8-12.
  • G. Wendt, Left ideals in 1-primitive near-rings, Math. Pannon., 16(1) (2005), 145-151.