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A Note on Semiprime Rings with Torsionless Injective Envelope

Published online by Cambridge University Press:  20 November 2018

Efraim P. Armendariz
Efraim P. Armendariz*
Affiliation:
University of Southwestern Louisiana, Lafayette Louisiana
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Satyanarayana establishes in [6] that a semiprime right selfinjective ring with ACC on annihilator right ideals is semisimple Artinian, thereby extending a similar result of Koh [5] for prime rings. A theorem of Faith [3, Theorem 5.2], shows that the annihilator chain condition on either side implies that a right selfinjective semiprime ring is semisimple Artinian. Noting that any selfinjective ring has torsionless injective envelope we consider the possibility of replacing selfinjectivity by torsionless together with an annihilator condition.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 16 , Issue 3 , September 1973 , pp. 429 - 431
Copyright
Copyright © Canadian Mathematical Society 1973

References

1. Armendariz, E. P., On finite-dimensional torsion-free modules and rings, Proc. Amer. Math. Soc. 24 (1970), 566–571.Google Scholar
2. Bass, H., Finitistic dimension and a homological generalization of semiprimary rings, Trans. Amer. Math. Soc. 95 (1960), 466–488.Google Scholar
3. Faith, C., Rings with ascending condition on annihilators, Nagoya Math. J. 27 (1966), 177–191.Google Scholar
4. Faith, C., Lectures on injective modules and quotient rings. Lecture Notes in Math. no. 49, Springer-Verlag, Berlin-New York, 1967.Google Scholar
5. Koh, K., A note on a self-injective ring, Canad. Math. Bull. 8 (1965), 29–32.Google Scholar
6. Satyanarayana, M., A note on a self-injective ring, Canad. Math. Bull. 14 (1971), 271–272.Google Scholar
7. Zelmanowitz, J., Endomorphism rings of torsionless modules, J. Algebra 5 (1967), 325–341.Google Scholar