Hostname: page-component-745bb68f8f-grxwn Total loading time: 0 Render date: 2025-01-13T07:40:50.419Z Has data issue: false hasContentIssue false
  • English
  • Français

On Division Near-Rings

Published online by Cambridge University Press:  20 November 2018

Steve Ligh
Steve Ligh*
Affiliation:
Texas A & M University, College Station, Texas
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.

The following results (9, Exercise 26, p. 10; 1, Theorem 9.2; 8, Theorem III. 1.11) are known.

(A) Let R be a ring with more than one element. Then R is a division ring ifand only if for every a ≠0 in R, there exists a unique b in R such that aba = a.

(B) Let R be a near-ring which contains a right identity e ≠ 0. Then R is adivision near-ring if and only if it contains no proper R-subgroups.

(C) Let R be a finite near-ring with identity. Then R is a division near-ringif and only if the R-module R+ is simple.

In this paper we will show that (A) can be generalized to distributively generated near-rings. We also will extend (B) and (C) to a larger class of near-rings.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 1366 - 1371
Copyright
Copyright © Canadian Mathematical Society 1969

References

1. Beidleman, J. C., On near-rings and near-ring modules, Doctoral Thesis, The Pennsylvania State University, 1964.Google Scholar
2. Clay, J. R. and Malone, J. J., Jr., The near-rings with identities on certain finite groups, Math. Scand. 19 (1966), 146–150.Google Scholar
3. Dickson, L. E., On finite algebras, Nachr. Ges. Wiss. Gottingen (1905), 358–393.Google Scholar
4. Frôhlich, A., Distributively generated near-rings. I: Ideal theory, Proc. London Math. Soc. 8 (1958), 76–94.Google Scholar
5. Heatherly, H. E., Near-rings on certain groups (to appear).Google Scholar
6. Ligh, Steve, On distributively generated near-rings, Proc. Edinburgh Math. Soc. (to appear).Google Scholar
7. Malone, J. J., Jr., Near-rings with trivial multiplications, Amer. Math. Monthly 74 (1967), 1111–1112.Google Scholar
8. Maxon, C. J., On near-rings and near-ring modules, Doctoral Thesis, State University of New York at Buffalo, Buffalo, New York, 1967.Google Scholar
9. McCoy, N., Theory of rings (Macmillan, New York, 1964).Google Scholar
10. Neumann, B. H., On the commutativity of addition, J. London Math. Soc. 15 (1940), 203–208.Google Scholar
11. Zassenhaus, H., Über endlich Fastkorper, Abh. Math. Sem. Univ. Hamburg 11 (1936), 187–220.Google Scholar
11. Zemmer, J. L., Near-fields, planar and non-planar, Math. Student 82 (1964), 145–150.Google Scholar