Hostname: page-component-84b7d79bbc-g7rbq Total loading time: 0 Render date: 2024-08-02T05:57:02.916Z Has data issue: false hasContentIssue false
  • English
  • Français

A special class of near rings

Published online by Cambridge University Press:  09 April 2009

Steve Ligh
Steve Ligh
Affiliation:
University of Southwestern Lousiana, Lafayette, Lousiana 70501, U. S. A.
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A near ring is a triple (R, +, · ) such that (R, + ) is a group, (R, ·) is a semigroup, and · is left distributive over +; i.e. w(x + z) = wx + wz for each w, x, z in R. A normal subgroup K of a near ring R is an ideal if (i) (m + k)n − mn is in K for all m, n in R and k in K, and (ii) RK ⊆ K. In particular, kernels of near ring homomorphisms are ideals. For various other definitions and elementary facts about near rings, see [5,8]. For each x in a near ring R, let A(x) = {y ∈ R: xy = 0}. A survey on several recent papers on near rings [2,3,6,7,8] shows that the concept of A(x) being an ideal was the main technique. The purpose of this note is to initiate a study of near rings having the property that each A(x) is an ideal.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 18 , Issue 4 , December 1974 , pp. 464 - 467
Copyright
Copyright © Australian Mathematical Society 1974

References

[1]Bell, H. E., ‘Duo rings: some applications to commutativity theorems’, Canad. Math. Bull. 11 (1968), 375–380.CrossRefGoogle Scholar
[2]Bell, H. E., ‘Near rings in which each element is a power of itself’, Bull. Austral. Math. Soc. 2 (1970), 363–368.CrossRefGoogle Scholar
[3]Bell, H. E., ‘Certain near rings are rings’, J. London Math. Soc. (2), 4 (1971), 264–270.CrossRefGoogle Scholar
[4]Ligh, S., ‘On division near rings’, Canad. J. Math. 21 (1969), 1366–1371.CrossRefGoogle Scholar
[5]Ligh, S., ‘On boolean near rings’, Bull. Austral. Math. Soc. 1 (1969), 375–279.CrossRefGoogle Scholar
[6]Ligh, S., ‘On regular near rings’, Math. Japan. 15 (1970), 7–13.Google Scholar
[7]Ligh, S., ‘On the commutativity of near rings, III’, Bull. Austral. Math. Soc. 6 (1972), 459–464.CrossRefGoogle Scholar
[8]Ligh, S., ‘The structure of a special class of near rings’, J. Austral. Math. Soc. 13 (1972), 141–146.CrossRefGoogle Scholar
[9]Ligh, S., On the additive group of finite near integral domains and simple d.g.near rings, Monatsh. Math. 76 (1972), 317–322.CrossRefGoogle Scholar
[10]Szeto, G., On a class of near rings, J. Austral. Math. Soc. 14 (1972), 17–19.CrossRefGoogle Scholar
[11]Thierrin, G., On duo rings, Canad. Math. Bull. 3 (1960), 167–172.CrossRefGoogle Scholar