Hostname: page-component-77c89778f8-vsgnj Total loading time: 0 Render date: 2024-07-20T13:01:48.260Z Has data issue: false hasContentIssue false
  • English
  • Français

On pseudo-distributive near-rings

Published online by Cambridge University Press:  20 January 2009

Gordon Mason
Gordon Mason
Affiliation:
Department of Mathematics & StatisticsUniversity of New BrunswickFredericton, N. B., Canada
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.

If G is a group and N a ring, the elements of the group ring NG can be thought of either as formal sums or as functions Φ:G→Nwith finite support. If N is a nearring, problems arise in trying to construct a group near-ring either way. In the first case, Meldrum [7] was abl to exploit properties of distributively generated near-rings (N, S) to build free (N,S)-products and hence a near-ring analogue of a group ring. For the latter case, Heatherly and Ligh [3] observed that the set of functions could be made into a near-ring under multiplication given by provided N satisfies

for all ai,bin∈N and k∈Z+. Such near-rings are called pseudo-distributive. In fact these are precisely the conditions under which the set Nk of k x k matrices over N is also a near-ring and then both NG and Nk are pseudo-distributive.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 28 , Issue 2 , June 1985 , pp. 133 - 141
Copyright
Copyright © Edinburgh Mathematical Society 1985

References

REFERENCES

1. Connell, G., On the group ring, Canad. J. Math. 15 (1963), 650–685.CrossRefGoogle Scholar
2. Heatherly, H., Distributive near-rings, Quart, J. Math. Ser. (2) 24 (1973), 63–70.CrossRefGoogle Scholar
3. Heatherly, H. and Ligh, S., Pseudo-distributive near-rings, Bull. Aust. Math. Soc. 12 (1975), 449–456.CrossRefGoogle Scholar
4. Ligh, S., A note on matrix near-rings, J. London Math. Soc. (2) 11 (1975), 383–384.CrossRefGoogle Scholar
5. Mason, G., Solvable and nilpotent near-ring modules, Proc. Amer. Math. Soc. 40 (1973), 351–357.CrossRefGoogle Scholar
6. Maxson, C., On near-rings and near-ring modules (Doctoral dissertation, SUNY at Buffalo, 1967).Google Scholar
7. Meldrum, J., The group distributively generated near-ring, Proc. London Math. Soc. (3) 32 (1976), 323–346.CrossRefGoogle Scholar
8. Passman, D., Infinite group rings (Marcel Dekker, New York, 1971).Google Scholar
9. Pilz, G., Near-rings (North Holland, Amsterdam, 1977).Google Scholar