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Cotorsion radicals and projective modules

Published online by Cambridge University Press:  17 April 2009

John A. Beachy
John A. Beachy
Affiliation:
Northern Illinois University, DeKalb, Illinois, USA.
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Abstract

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We study the notion (for categories of modules) dual to that of torsion radical and its connections with projective modules.

Torsion radicals in categories of modules have been studied extensively in connection with quotient categories and rings of quotients. (See [8], [12] and [13].) In this paper we consider the dual notion, which we have called a cotorsion radical. We show that the cotorsion radicals of the category RM correspond to the idempotent ideals of R. Thus they also correspond to TTF classes in the sense of Jans [9].

It is well-known that the trace ideal of a projective module is idempotent. We show that this is in fact a consequence of the natural way in which every projective module determines a cotorsion radical. As an application of these techniques we study a question raised by Endo [7], to characterize rings with the property that every finitely generated, projective and faithful left module is completely faithful. We prove that for a left perfect ring this is equivalent to being an S-ring in the sense of Kasch. This extends the similar result of Morita [15] for artinian rings.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 5 , Issue 2 , October 1971 , pp. 241 - 253
Copyright
Copyright © Australian Mathematical Society 1971

References

[0]Alin, J.S. and Armendariz, E.P., “TTF-dasses over perfect rings”, J. Austral. Math. Soc. 11 (1970), 499–503.CrossRefGoogle Scholar
[1]Auslander, Maurice and Goldman, Oscar, “Maximal orders”, Trans. Amer. Math. Soc. 97 (1960), 1–24.CrossRefGoogle Scholar
[2]Azumaya, Goro, “Completely faithful modules and self-injective rings”, Nagoya Math. J. 27 (1966), 697–708.Google Scholar
[3]Bass, Hyman, “Finitistic dimension and a homological generalization of semi-primary rings”, Trans. Amer. Math. Soc. 95 (1960), 466–488.CrossRefGoogle Scholar
[4]Beachy, John A., “Generating and cogenerating structures”, Trans. Amer. Math. Soc. 158 (1971), (to appear).Google Scholar
[5]Dlab, Vlastimil, “A characterization of perfect rings”, Pacific J. Math. 33 (1970), 79–88.Google Scholar
[6]Eisenbud, David and Robson, J.C., “Hereditary Noetherian prime rings”, J. Algebra 16 (1970), 86–104.CrossRefGoogle Scholar
[7]Endo, Shizuo, “Completely faithful modules and quasi-Frobenius algebras”, J. Math. Soc. Japan 19 (1967), 437–456.CrossRefGoogle Scholar
[8]Goldman, Oscar, “Rings and modules of quotients”, J. Algebra 13 (1969), 10–47.CrossRefGoogle Scholar
[9]Jans, J.P., “Some aspects of torsion’, Pacific J. Math. 15 (1965), 1249–1259.CrossRefGoogle Scholar
[10]Kurata, Yoshiki, “On an n-fold torsion theory in the category RM”, J. Algebra (to appear).Google Scholar
[11]Lambek, J., Lectures on rings and modules (Blaisdell, Waltham, Massachusetts; 1966).Google Scholar
[12]Lambek, Joachim, Torsion theories, additive semantics, and rings of quotients (Lecture Notes in Mathematics, No. 177. Springer-Verlag, Berlin, Heidelberg, New York, 1971).Google Scholar
[13]Maranda, J.-M., “Injective structures”, Trans. Amer. Math. Soc. 110 (1964), 98–135.Google Scholar
[14]Michler, Gerhard, “Idempotent ideals in perfect rings”, Canad. J. Math. 21 (1969), 301–309.CrossRefGoogle Scholar
[15]Morita, Kiiti, “On S-rings in the sense of F. Kasch”, Nagoya J. Math. 27 (1966), 687–695.CrossRefGoogle Scholar