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Faithfully Exact Functors and Their Applications to Projective Modules and Injective Modules

Published online by Cambridge University Press:  22 January 2016

Takeshi Ishikawa
Takeshi Ishikawa*
Affiliation:
Tokyo Metropolitan University
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The aim of this paper is to study a property of a special kind of exact functors and give some applications to projective modules and injective modules.

In section 1 we introduce the notion of faithfully exact functors [Definition 1] as a generalization of the functor T(X) = X⊗M, where M is a faithfully flat module, and give a property of this class of functors [Theorem 1.1]. Next, applying this general theory to functors ⊗ and Horn, we define the notion of faithfully projective modules [Definition 2] and faithfully injective modules [Definition 3]. In the commutative case “faithfully projective” means, however, simply “projective and faithfully flat” [Proposition 2.3]. In section 2, equivalent conditions for a projective module P to be faithfully projective are given [Theorem 2.2, Proposition 2. 3 and 2.4]. And a simpler proof is given to Y. Hinohara’s result [6] asserting that projective modules over an indecomposable weakly noetherian ring are faithfully flat [Proposition 2.5]. In section 3, we consider faithfully injective modules.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 24 , June 1964 , pp. 29 - 42
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1964

References

[1] Azumaya, G., A duality theory for injective modules, Amer. J. of Math., 81 (1960), pp. 249–278.Google Scholar
[2] Bass, H., Injective dimension in Noetherian rings. Trans. Amer. Math. Soc, 102 (1962), pp. 18–29.Google Scholar
[3] Bourbaki, N., Algèbre commutative, Chap. 1, (Hermann Paris, 1961).Google Scholar
[4] Cartan, H. and Eilenberg, S., Homological algebra, (Princeton University Press, 1956).Google Scholar
[5] Goldman, O., Determinants in projective modules, Nagoya Math. J., 18 (1961), pp. 27–36.Google Scholar
[6] Hinohara, Y., Projective modules over weakly Noetherian rings, J. of Math. Soc. of Japan, Vol. 15, No. 1 (1963), pp. 75–88.Google Scholar
[7] Kaplansky, I., Homological dimension of rings and modules, Univ. of Chicago (1959), (mimeographed notes).Google Scholar
[8] Matlis, E., Injective modules over Noetherian rings, Pacific J. of Math., 8 (1958), pp. 511–528.Google Scholar
[9] Matlis, E., Some properties of Noetherian domain of dimension one, Canadian J. of Math. 13 (1961), pp. 569–586.Google Scholar
[10] Northcott, D. G., An introduction to homological algebra, (Cambridge University Press, 1960),Google Scholar