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A Simple Proof of a Theorem on Reduced Rings

Published online by Cambridge University Press:  20 November 2018

Abraham A. Klein
Abraham A. Klein*
Affiliation:
Department of Mathematical Sciences, Tel-Aviv University, Tel-Aviv, Israel
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We give a simple proof of a theorem by Andrunakievič and Rjabuhin which states that a reduced ring is a subdirect product of entire rings. Our proof makes no use of m-systems and is in some sense similar to the proof of the corresponding theorem in the commutative case due to Krull.

A reduced ring is a ring without non-zero nilpotent elements. It is wellknown that if a reduced ring is commutative, then it is a subdirect product of integral domains [2]. This result has been generalized to arbitrary reduced rings [1]. The proof in the general case is somewhat complicated. We present a simple proposition which leads to a simple proof of the general case.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 23 , Issue 4 , 01 December 1980 , pp. 495 - 496
Copyright
Copyright © Canadian Mathematical Society 1980

References

1. Andrunakievic, V. A. and Rjabuhin, Ju. M., Rings without nilpotent elements and completely simple rings, Soviet Math. Dokl. 9 (1968), 565-567, MR 37 #6320.Google Scholar
2. Krull, W., Idealtheorie in Ringen ohne Endlichkeitsbedingung, Math. Ann. 101 (1929), 729-744.Google Scholar