Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-28T05:05:02.506Z Has data issue: false hasContentIssue false
  • English
  • Français

Characterizations of a two dimensional Euclidean ring among near-rings

Part of: Generalizations

Published online by Cambridge University Press:  26 February 2010

K. D. Magill Jr
K. D. Magill Jr
Affiliation:
Department of Mathematics. College of Arts and Sciences, Mathematics Building, Room 244, SUNY at Buffalo, Buffalo, NY 14260-2900, U.S.A E-mail:[email protected]
Get access

Abstract

All those multiplications on the two-dimensional Euclidean group are determined such that the resulting non-associative topological nearring has (1, 0) for a left identity and has the additional property that every element of the near-ring is a right divisor of zero. This result, together with several previous results, is then used to show that any one of several common algebraic properties is sufficient to characterize one particular two-dimensional Euclidean ring within the class of all two dimensional Euclidean near-rings. Specifically, it is proved that, if N is a topological near-ring with a left identity whose additive group is the two-dimensional Euclidean group, then the following assertions are equivalent: (1) the left identity is not a right identity, (2) N contains a non-zero left annihilator, (3) every element of N is a right divisor of zero, (4) Nw≠N for all w∈N, (5) N is isomorphic to the topological ring whose additive group is the two dimensional Euclidean group and whose multiplication is given by (v1, V2)(w1W2) = (v1w1, v1w2).

MSC classification

Secondary: 16Y30: Near-rings
Type
Research Article
Information
Mathematika , Volume 49 , Issue 1-2 , June 2002 , pp. 59 - 64
Copyright
Copyright © University College London 2002

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Clay, J. R.. Near-rings: Geneses and applications (Oxford University Press, New York, 1992).CrossRefGoogle Scholar
2.Heuer, G. A.. Continuous multiplications in R 2. Math. Mag., 45 (1972) 72–77.CrossRefGoogle Scholar
3.Magill, K. D. Jr. Topological near-rings whose additive groups are Euclidean. Monatin. Math., 119 (1995)281–301.CrossRefGoogle Scholar
4.Magill, K. D. Jr. The topological near-ring on the Euclidean plane which has a left identity which is not a right identity. Semigroup Forum, 57 (1998) 435–437.CrossRefGoogle Scholar
5.Magill, K. D. Jr. Euclidean near-rings with a left identity and a nonzero nilpotent element. Alg. Coll.. 6(2) (1999) 133–143.Google Scholar
6.Magill, K. D. Jr. Euclidean near-rings with a proper nonzero closed connected right ideal and a left zero not in that ideal. Southeast Asian Bull. Math., 23 (1999) 79–109.Google Scholar
7.Magill, K. D. Jr. Some conditions which force Euclidean near-rings to be rings. Demonslratio Math., 34. (2001) 51–58.Google Scholar
8.Magill, K. D. Jr. Linear right ideal near-rings. Internal. J. Math. Math. Sci., 27 (2001), 663–674.CrossRefGoogle Scholar
9.Meldrum, J. D. P., Near-rings and their links with groups. Pitman Research Notes, Vol. 134 (Pitman, London, 1985).Google Scholar
10.Pilz, G.. Near-rings. North Holland Math. Studies, Vol. 23, revised ed. (North Holland, Amsterdam. 1983).Google Scholar