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A generalization of a theorem of Wedderburn

Published online by Cambridge University Press:  17 April 2009

Steve Ligh
Steve Ligh
Affiliation:
Department of Mathematics, University of Southwestern Louisiana, Lafayette, Louisiana, USA.
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Abstract

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Outcalt and Yaqub have extended the Wedderburn Theorem which states that a finite division ring is a field to the case where R is a ring with identity in which every element is either nilpotent or a unit. In this paper we generalize their result to the case where R has a left identity and the set of nilpotent elements is an ideal. We also construct a class of non-commutative rings showing that our generalization of Outcalt and Yaqub's result is real.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 8 , Issue 2 , April 1973 , pp. 181 - 185
Copyright
Copyright © Australian Mathematical Society 1973

References

[1]Herstein, I.N., “A note on rings with central nilpotent elements”, Proc. Amer. Math. Soc. 5 (1954), 620.CrossRefGoogle Scholar
[2]Jacobson, Nathan, Lectures in abstract algebra, Volume I (Van Nostrand, New York, Toronto, London, 1951).Google Scholar
[3]Johnsen, E.C., Outcalt, D.L. and Yaqub, Adil, “An elementary commutativity theorem for rings”, Amer. Math. Monthly 75 (1968), 288–289.CrossRefGoogle Scholar
[4]Luh, J., “A commutativity theorem for primary rings”, Acta Math. Acad. Sci. Hungar. 22 (1971), 211–213.CrossRefGoogle Scholar
[5]Malone, J.J. Jr, “Near-rings with trivial multiplications”, Amer. Math. Monthly 74 (1967), 1111–1112.CrossRefGoogle Scholar
[6]Moore, H.G., “On commutativity in certain rings”, Bull. Austral. Math. Soc. 2 (1970), 107–115.CrossRefGoogle Scholar
[7]Outcalt, D.L. and Yaqub, Adil, “A generalization of Wedderburn's Theorem”, Proc. Amer. Math. Soc. 18 (1967), 175–177.Google Scholar
[8]Outcalt, D.L. and Yaqub, Adil, “A commutativity theorem for rings”, Bull. Austral. Math. Soc. 2 (1970), 95–99.CrossRefGoogle Scholar