Taiwanese Journal of Mathematics

Minimal Ideals and Primitivity in Near-rings

Gerhard Wendt

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We address and answer the question when a minimal ideal of a zero symmetric near-ring is a primitive near-ring. This implies that a minimal ideal of a zero symmetric near-ring is a simple near-ring in many natural situations.

Article information

Taiwanese J. Math., Volume 23, Number 4 (2019), 799-820.

Received: 27 August 2018
Revised: 4 December 2018
Accepted: 10 December 2018
First available in Project Euclid: 18 July 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

simple near-rings primitive near-rings left ideals minimal ideals subdirect irreducibility


Wendt, Gerhard. Minimal Ideals and Primitivity in Near-rings. Taiwanese J. Math. 23 (2019), no. 4, 799--820. doi:10.11650/tjm/181206. https://projecteuclid.org/euclid.twjm/1563436869

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  • G. Birkenmeier and H. Heatherly, Minimal ideals in near-rings, Comm. Algebra 20 (1992), no. 2, 457–468.
  • C. C. Ferrero and G. Ferrero, Nearrings: Some Developments Linked to Semigroups and Groups, Advances in Mathematics (Dordrecht) 4, Kluwer Academic Publishers, Dordrecht, 2002.
  • K. Kaarli, Minimal ideals in near-rings, Tartu Riikl. Ül. Toimetised Vih. 366 (1975), 105–142.
  • ––––, On Jacobson type radicals of near-rings, Acta Math. Hungar. 50 (1987), no. 1-2, 71–78.
  • ––––, On minimal ideals of distributively generated near-rings, in: Contributions to General Algebra, 7 (Vienna, 1990), 201–204, Hölder-Pichler-Tempsky, Vienna, 1991.
  • J. D. P. Meldrum, Near-rings and their links with groups, Research Notes in Mathematics 134, Pitman (Advanced Publishing Program), Boston, MA, 1985.
  • G. Pilz, Near-rings: The Theory and its Applications, Second edition, North-Holland Mathematics Studies 23, North-Holland Publishing Co., Amsterdam, 1983.
  • G. Wendt, Left ideals in $1$-primitive near-rings, Math. Pannon. 16 (2005), no. 1, 145–151.
  • ––––, $1$-primitive near-rings, Math. Pannon. 24 (2013), no. 2, 269–287.
  • ––––, $0$-primitve near-rings, minimal ideals and simple near-rings, Taiwanese J. Math. 19 (2015), no. 3, 875–905.