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Characterizations of Modules

Published online by Cambridge University Press:  20 November 2018

David J. Fieldhouse
David J. Fieldhouse*
Affiliation:
University of Guelph, Guelph, Ontario
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In this paper we use the Bourbaki [2] conventions for rings and modules. All rings are associative but not necessarily commutative and have a 1; all modules are unital.

Bass [1] calls a ring A left perfect if and only if every left A -module has a projective cover, which he shows is equivalent to every flat left A -module being projective. Bass calls a ring A semi-perfect if and only if every finitely generated module has a projective cover and shows that this concept is leftright symmetric.

We will define a ring A to be quasi-perfect if and only if every finitely generated flat left A -module is projective.

An exercise [6, Exercise 10, p. 136] is given by Lambek to show that every semi-perfect ring is quasi-perfect.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 23 , Issue 4 , August 1971 , pp. 608 - 610
Copyright
Copyright © Canadian Mathematical Society 1971

References

1. Bass, H., Finitistic hornological dimension and a homological generalization of semi-primary rings, Trans. Amer. Math. Soc. 95 (1960), 466–488.Google Scholar
2. Bourbaki, N., Algèbre commutative, Chap. I (Hermann, Paris, 1961).Google Scholar
3. Cohn, P. M., On the free product of associative rings, Math. Z. 71 (1959), 380–398.Google Scholar
4. Fieldhouse, D., Pure theories, Math. Ann. 184 (1969), 1–18.Google Scholar
5. Fieldhouse, D., Pure simple and indecomposable rings, Can. Math. Bull. 13 (1970), 71–78.Google Scholar
6. Lambek, J., Lectures on rings and modules (Blaisdell, Waltham, Mass., 1966).Google Scholar