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# Characterizations of a two dimensional Euclidean ring among near-rings

## 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 wN, (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).

## References

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1.Clay, J. R.. Near-rings: Geneses and applications (Oxford University Press, New York, 1992).
2.Heuer, G. A.. Continuous multiplications in R 2. Math. Mag., 45 (1972) 7277.
3.Magill, K. D. Jr. Topological near-rings whose additive groups are Euclidean. Monatin. Math., 119 (1995)281301.
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) 435437.
5.Magill, K. D. Jr. Euclidean near-rings with a left identity and a nonzero nilpotent element. Alg. Coll.. 6(2) (1999) 133143.
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) 79109.
7.Magill, K. D. Jr. Some conditions which force Euclidean near-rings to be rings. Demonslratio Math., 34. (2001) 5158.
8.Magill, K. D. Jr. Linear right ideal near-rings. Internal. J. Math. Math. Sci., 27 (2001), 663674.
9.Meldrum, J. D. P., Near-rings and their links with groups. Pitman Research Notes, Vol. 134 (Pitman, London, 1985).
10.Pilz, G.. Near-rings. North Holland Math. Studies, Vol. 23, revised ed. (North Holland, Amsterdam. 1983).
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