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Rings with Ascending Condition on Annihilators

Published online by Cambridge University Press:  22 January 2016

Carl Faith
Carl Faith*
Affiliation:
Institute for Defense Analyses Princeton, New Jersey
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Quasi-frobenius (= QF) rings have many interesting characterizations. One such, due to Ikeda [17] is that these rings are right (left) artinian and right (left) self-injective. Thus, if R is QF, then R is right (left) noetherian, so each direct sum of injective right R-modules is injective: in particular, each free, hence, each projective, R-module is injective. One object of this paper is to report that this property characterizes QF-rings:

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 27 , Issue 1 , February 1966 , pp. 179 - 191
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1966

References

[1] Baer, Rheinhold, Abelian groups which are direct summands of every containing abellan group, Bull. Amer. Math. Soc. 46 (1940), 800–806.Google Scholar
[2] Bass, Hyman, Finitistic dimension and a homological generalization of semi-primary rings, Trans. Amer. Math. Soc. 95 (1960), 466–488.Google Scholar
[3] Chase, Stephen U., Ph. D. Thesis, University of Chicago.Google Scholar
[4] Eckmann, Beno and Schopf, A., Über Injektive Moduln, Archiv der Math. 4 (1953), 74–78.CrossRefGoogle Scholar
[5] Eilenberg, Samuel and Nakayama, Tadasi, On the dimension of modules and algebras, II (Frobenius algebras and quasi-frobenius rings) Nagoya Math. J. 9 (1966), 1–16.Google Scholar
[6] Faith, Carl, Lectures on injective modules and quotient rings, (revised edition), Springer, Heidelberg, 1966.Google Scholar
[7] Faith, Carl, and Walker, Elbert A., Direct sum representations of injective modules, J. Algebra (to appear).Google Scholar
[8] Goldie, Alfred W., The structure of prime rings under ascending chain conditions, Proc. London Math. Soc. (3) 8 (1958), 589–608.CrossRefGoogle Scholar
[9] Goldie, Alfred W., Semiprime rings with maximum condition, Proc. London Math. Soc. (3) 10 (1960), 201–220.CrossRefGoogle Scholar
[10] Hopkins, Charles, Rings with minimal condition for left ideals, Ann. of Math. 40 (1939), 712–730.CrossRefGoogle Scholar
[11] Ikeda, Masatosi and Nakayama, Tadasi, On some characteristic properties of quasi-frobenius and regular rings, Proc. Amer. Math. Soc. 5 (1954), 15–19.Google Scholar
[12] Johnson, Richard E., The extended centralizer of a ring over a module, Proc. Amer. Math. Soc. 2 (1951), 891–895.CrossRefGoogle Scholar
[13] Utumi, Yuzo, On a theorem on modular lattices, Proc. Japan Academy, 35 (1959), 16–21.Google Scholar
[14] Neumann, John von, On regular rings, Proc. Nat. Acad. Sci. USA 22 (1936); 707–713.Google Scholar
[15] Walker, Carol P. and Walker, Elbert A., Quotient categories of modules, Trans. Amer. Math. Soc. (to appear).Google Scholar
[16] Wong, Edward T. and Johnson, Richard E., Self-injective rings, Can. Math. Bull. 2 (1959), 167–173.CrossRefGoogle Scholar
[17] Ikeda, Masatoshi, A characterization of quasi-frobenius rings, Osaka Math, J. 4 (1952), 203–210.Google Scholar