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On (von Neumann) regular rings

Published online by Cambridge University Press:  20 January 2009

R. Yue Chi Ming
R. Yue Chi Ming
Affiliation:
Universite Paris VII, U.E.R. de Mathematiques, Tour 45-55, 2, Place Jussieu, 75005 Paris
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Throughout, A denotes an associative ring with identity and “module” means “left, unitary A-module”. In (3), it is proved that A is semi-simple, Artinian if A is a semi-prime ring such that every left ideal is a left annihilator. A natural question is whether a similar result holds for a (von Neumann) regular ring. The first proposition of this short note is that if A contains no non-zero nilpotent element, then A is regular iff every principal left ideal is the left annihilator of an element of A. It is well-known that a commutative ring is regular iff every simple module is injective (I. Kaplansky, see (2, p. 130)). The second proposition here is a partial generalisation of that result.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 19 , Issue 1 , March 1974 , pp. 89 - 91
Copyright
Copyright © Edinburgh Mathematical Society 1974

References

REFERENCES

(1) Cozzens, J. H., Homological properties of the ring of differential polynomials, Bull. Amer. Math. Soc. 76 (1970), 75–79.CrossRefGoogle Scholar
(2) Faith, C., Lectures on injective modules and quotient rings. Lecture notes in Mathematics N° 49 (Springer-Verlag, Berlin-Heidelberg-New York, 1967).CrossRefGoogle Scholar
(3) Ming, R. Yue Chi, On elemental annihilator rings, Proc. Edinburgh Math. Soc. 17 (1970), 187–188.CrossRefGoogle Scholar