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On Continuous Regular Rings and Semisimple Self Injective Rings

Published online by Cambridge University Press:  20 November 2018

Yuzo Utumi
Yuzo Utumi*
Affiliation:
Osaka Women's University and McGill University
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Brainerd and Lambek (2, Corollary 4) have proved recently that any complete Boolean ring is self-injective. It is easy to see that every complete Boolean ring is a continuous regular ring, that is, a regular ring of which the lattice of principal left ideals is continuous. This suggests that in a continuous regular ring it might be possible to prove the injectivity. However, a simple example (Example 3) shows that the conjecture is not true in general. Our main theorem is the following. Every continuous regular ring with no ideals of index 1 is (both left and right) self-injective (Theorem 3).

It is known to Wolfson (13, Theorem 5.1) and Zelinsky (15) that the ring S of all linear transformations of a vector space of dimension ≥ 2 over a division ring is generated by idempotents and also by non-singular elements.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 12 , 1960 , pp. 597 - 605
Copyright
Copyright © Canadian Mathematical Society 1960

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