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On distributively generated near-rings1

Published online by Cambridge University Press:  20 January 2009

Steve Lich
Steve Lich
Affiliation:
Department of Mathematics, Texas A ' M UniversityCollege Station, Texas 77843U.S.A.
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The following theorems in ring theory are well-known:

1. Let R be a ring. If e is a unique left identity, then e is also a right identity.

2. If R is a ring with more than one element such that aR = R for every nonzero element a ε R, then R is a division ring.

3. A ring R with identity e ≠ 0 is a division ring if and only if it has no proper right ideals.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 16 , Issue 3 , June 1969 , pp. 239 - 243
Copyright
Copyright © Edinburgh Mathematical Society 1969

References

REFERENCES

(1)Frohlich, A.Distributively generated near-rings (I. Ideal Theory), Proc. London Math. Soc. (3) 8 (1958), 76–94.CrossRefGoogle Scholar
(2)Malone, J. J. Jr.Near-rings with trivial multiplications, Amer. Math. Monthly, 74 (1967), 1111–1112.CrossRefGoogle Scholar
(3)Neumann, B. H.On the commutativity of addition, J. London Math. Soc. 15 (1940), 203–208.CrossRefGoogle Scholar
(4)Zassenhausy, H.Über endlich Fastkorper, Abh. Math. Sem. Univ. Hamburg, 11 (1936), 187–220.CrossRefGoogle Scholar