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Imbedding a Regular Ring in a Regular Ring with Identity

Published online by Cambridge University Press:  22 January 2016

Nenosuke Funayama
Nenosuke Funayama*
Affiliation:
Yamagata University
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In [1] L. Fuchs and I. Halperin have proved that a regular ring R is isomorphic to a two-sided ideal of a regular ring with identity. ([1] Theorem 1). Their methed is to imbed the regular ring R in the ring of all pairs (a, p) with a ∊ R and p from a suitable commutative regular ring S with identity such that R is an algebra over S. Thus S may be seen as the ring of R — R endomorphisms of the additive group of R. The following question is naturally raised: Is it true that the ring of all R-R endomorphisms of a rugular ring is a commutative regular ring? The main purpose of this paper is to answer this question affirmatively. (Theorem 1). After established this theorem we can follow the method in [1] to solve the problem in the title.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 27 , Issue 1 , February 1966 , pp. 61 - 64
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1966

References

[1] Fuchs, L. and Halparin, I., On the embedding of a regular ring in a regular ring with identity, Fundamenta Mathematicae LIV (1964), pp. 287–290.Google Scholar