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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
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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

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