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On radical extensions of rings

Published online by Cambridge University Press:  09 April 2009

Efraim P. Armendariz
Efraim P. Armendariz
Affiliation:
University of Nebraska Lincoln, Nebraska, U.S.A.
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A ring K is a radical extension of a subring B if for each x ∈ K there is aninteger n = n(x) > 0 such that xn ∈ B. In [2] and [3], C. Faith considered radical extensions in connection with commutativity questions, as well as the generation of rings. In this paper additional commutativity theorems are established, and rings with right minimum condition are examined. The main tool is Theorem 1.1 which relates the Jacobson radical of K to that of B, and is of independent interest in itself. The author is indebted to the referee for his helpful suggestions, in particular for the strengthening of Theorem 2.1.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 7 , Issue 4 , November 1967 , pp. 552 - 554
Copyright
Copyright © Australian Mathematical Society 1967

References

[1]Faith, Carl, ‘Submodules of rings’, Proc. Amer. Math. Soc. 10 (1959), 596–606.CrossRefGoogle Scholar
[2]Faith, Carl, ‘Algebraic division ring extensions’, Proc. Amer. Math. Soc. 11 (1960), 43–53.CrossRefGoogle Scholar
[3]Faith, Carl, ‘Radical extensions of rings’, Proc. Amer. Math. Soc. 12 (1961), 274–283.CrossRefGoogle Scholar
[4]Herstein, I. N., ‘A theorem on rings’, Canadian J. Math. 5 (1953), 238–241.CrossRefGoogle Scholar
[5]McCoy, Neal H., Theory of Rings (MacMillan, New York 1964).Google Scholar