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Note on P.P. Rings: (A Supplement to Hattori’s Paper)

Published online by Cambridge University Press:  22 January 2016

Shizuo Endo
Shizuo Endo*
Affiliation:
Kanto Gakuin University
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A ring R is called, according to [2], a left p.p. ring if any principal left ideal of R is projective. A ring which is left and right p.p. is called a p.p. ring.

In this short note we shall give some additional remarks to A. Hattori [2]. In Proposition 1 we shall give a characterization of commutative p.p. rings, and in Proposition 3 we shall give a generalization of Proposition 17 and 18 in [2], which shows also that the modified torsion theory over commutative p.p. rings coincides with the usual torsion theory.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 17 , August 1960 , pp. 167 - 170
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1960

References

[1] Cartan, H. and Eilenberg, S., Homological Algebra, Princeton Univ. Press, 1956.Google Scholar
[2] Hattori, A., A foundation of torsion theory for modules over general rings, Nagoya Math. Jour., this issue.Google Scholar
[3] Akizuki, Y., The theory of local rings, Lecture notes at Univ. of Chicago, 1958,Google Scholar
[4] Endo, S., Regular rings and semi-hereditary rings, to appear.Google Scholar