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Elementary proofs of a theorem of Wedderburn and a theorem of Jacobson

Elementary proofs of a theorem of Wedderburn and a theorem of Jacobson Elementary Proofs of a Theorem of Vdedderburn and a Theorem of Jacobson By TAKASI NAGAH.~RA and HIsAo TOMINAGA (Okayama) Recently, K. ROGERS [1] and J. W. W~sL~Y [2] gave elementary proofs of a theorem of Jacobsen. Their proofs are very similar and decid- edly due to a well-known theorem of Wedderburn. ([1] contains fairly complete references to the proofs of the theorem and some related results.) In this note, we shall give a self-contained proof of Jacobson's theorem, giving a new elementary proof of Wedderburn's theorem. Lemma. Every/inite ring without non-zero nilpotent elements is a direct sum o//inite/ields. Proof. Assume that there exists a finite ring A without non-zero nilpotent elements which does not coincide with its center C. Then, we may assume further that every proper subring of A is commutative. If x is an arbitrary non-zero element of A then x h+~ = x h for some positive integers h, k, and then x h~ is a non-zero central idempotent. Since A can not be a direct sum of non-zero ideals, x a~ must be the identity element 1 of A. Hence, x is a unit of A, namely, A is a division ring. Now, let a http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg Springer Journals

Elementary proofs of a theorem of Wedderburn and a theorem of Jacobson

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References (3)

Publisher
Springer Journals
Copyright
Copyright
Subject
Mathematics; Mathematics, general; Algebra; Differential Geometry; Number Theory; Topology; Geometry
ISSN
0025-5858
eISSN
1865-8784
DOI
10.1007/BF02993501
Publisher site
See Article on Publisher Site

Abstract

Elementary Proofs of a Theorem of Vdedderburn and a Theorem of Jacobson By TAKASI NAGAH.~RA and HIsAo TOMINAGA (Okayama) Recently, K. ROGERS [1] and J. W. W~sL~Y [2] gave elementary proofs of a theorem of Jacobsen. Their proofs are very similar and decid- edly due to a well-known theorem of Wedderburn. ([1] contains fairly complete references to the proofs of the theorem and some related results.) In this note, we shall give a self-contained proof of Jacobson's theorem, giving a new elementary proof of Wedderburn's theorem. Lemma. Every/inite ring without non-zero nilpotent elements is a direct sum o//inite/ields. Proof. Assume that there exists a finite ring A without non-zero nilpotent elements which does not coincide with its center C. Then, we may assume further that every proper subring of A is commutative. If x is an arbitrary non-zero element of A then x h+~ = x h for some positive integers h, k, and then x h~ is a non-zero central idempotent. Since A can not be a direct sum of non-zero ideals, x a~ must be the identity element 1 of A. Hence, x is a unit of A, namely, A is a division ring. Now, let a

Journal

Abhandlungen aus dem Mathematischen Seminar der Universität HamburgSpringer Journals

Published: Nov 18, 2008

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