Hostname: page-component-745bb68f8f-v2bm5 Total loading time: 0 Render date: 2025-01-27T22:20:56.887Z Has data issue: false hasContentIssue false
  • English
  • Français

On Left Bipotent Near-Rings

Published online by Cambridge University Press:  20 January 2009

J. L. Jat  and
S. C. Choudhary
J. L. Jat
Affiliation:
Department of Mathematics, SBSH, University of UdaipurUdaipur-313001, India
S. C. Choudhary
Affiliation:
Department of Mathematics, SBSH, University of UdaipurUdaipur-313001, India
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 N is defined to be left bipotent if Na = Na2 for each a in N. Many properties of such near-rings are proved in Section 1, and results of Chandran (4) are generalised. Most of the results are different from, and contrary to, the ring case. Necessary and sufficient conditions have also been obtained under which such near-rings become regular. Section 2 deals with left bipotent near-rings without zero divisors. Some structure theorems for direct sum decompositions and J(N) = (0) are proved and it is shown that for a left bipotent S-near-ring, the singular ‘set’ S(N) = 0. Necessary examples and counter examples are supplied.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 22 , Issue 2 , June 1979 , pp. 99 - 107
Copyright
Copyright © Edinburgh Mathematical Society 1979

References

REFERENCES

(1) Beidleman, J. C., A note on regular near rings, J. Indian Math. Soc. 33 (1969), 207–210.Google Scholar
(2) Beidleman, J. C., A radical for near ring modules, Mich. Math. J. 12 (1965), 377–383.CrossRefGoogle Scholar
(3) Blackett, D. W., Simple and semisimple near rings, Proc. Amer. Math. Soc. 4 (1953), 772–785.CrossRefGoogle Scholar
(4) Chandran, V. R., On right bipotent rings, Abstract in Notices Amer. Math. Soc. 11. (1970).Google Scholar
(5) Choudhary, S. C. and Tewari, K., On strictly semisimple near rings, Abh. Math. Sem. Univ. Hamburg 40 (1974), 256–264.CrossRefGoogle Scholar
(6) Clay, J. R., Imbedding an arbitrary ring in a nontrivial near ring, Amer. Math. Monthly 74 (1967), 406–407.CrossRefGoogle Scholar
(7) Clay, J. R., The near rings on groups of low order, Math. Z. 104 (1968), 364–371.CrossRefGoogle Scholar
(8) Fröhlich, A., Distributively generated near rings (I ideal Theory), Proc. London Math. Soc. 8 (1958), 76–94.CrossRefGoogle Scholar
(9) Heatherly, H. E., Near rings without nilpotent elements, Pub. Math. 12 20 (1973), 201–205.Google Scholar
(10) Ligh, S., On regular near rings, Math. Japonicae Ser (1) 15 (1970), 7–13.Google Scholar
(11) Maxson, C. J., On near rings and near ring modules (Ph.D. Dissertation, State Univ. of New York at Buffalo, New York, 1967).Google Scholar
(12) Neumann, B. H., On the commutativity of addition, J. London Math. Soc. 15 (1940), 203–208.CrossRefGoogle Scholar
(13) Oswald, A., Near rings in which every JV-subgroup is principal, Proc. London Math. Soc. 28 (1974), 68–88.Google Scholar
(14) Ramakotaiah, D., Radicals for near rings, Math. Z. 97 (1967), 45–56.CrossRefGoogle Scholar
(15) Szász, F., A class of regular rings, Monatsh. Math. 75 (1971), 168–172.CrossRefGoogle Scholar