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On quasi-duo rings

Published online by Cambridge University Press:  18 May 2009

Hua-Ping Yu
Hua-Ping Yu
Affiliation:
Department of Mathematics, The University of Iowa, Iowa City, IA 52242, U.S.A.
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Bass [1] proved that if R is a left perfect ring, then R contains no infinite sets of orthogonal idempotents and every nonzero left R-module has a maximal submodule, and asked if this property characterizes left perfect rings ([1], Remark (ii), p. 470). The fact that this is true for commutative rings was proved by Hamsher [12], and that this is not true in general was demonstrated by examples of Cozzens [7] and Koifman [14]. Hamsher's result for commutative rings has been extended to some noncommutative rings. Call a ring left duo if every left ideal is two-sided; Chandran [5] proved that Bass’ conjecture is true for left duo rings. Call a ring R weakly left duo if for every r ε R, there exists a natural number n(r) (depending on r) such that the principal left ideal Rrn(r) is two-sided. Recently, Xue [21] proved that Bass’ conjecture is still true for weakly left duo rings.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 37 , Issue 1 , January 1995 , pp. 21 - 31
Copyright
Copyright © Glasgow Mathematical Journal Trust 1995

References

REFERENCES

1.Bass, H., Finitistic dimension and a generalization of semiprimary rings, Trans. Amer. Math. Soc. 95 (1960), 466–488.CrossRefGoogle Scholar
2.Brown, S. H., Rings over which every simple module is rationally complete, Canad. J. Math. 25 (1973), 693–701.CrossRefGoogle Scholar
3.Burgess, W. D. and Stephenson, W., Rings all of whose Pierce stalks are local, Canad. Math. Bull. 22 (1979), 159–164.CrossRefGoogle Scholar
4.Crawley, P. and Jonsson, B., Refinements for infinite direct decompositions of algebraic systems, Pacific J. Math. 14 (1964), 797–855.CrossRefGoogle Scholar
5.Chandran, V. R., On a conjecture of Hyman Bass, Pure Appl. Math. Sci. 4 (1976), 125–131.Google Scholar
6.Courter, R. C., Finite dimensional right duo algebras are duo, Proc. Amer. Math. Soc. 84 (1982), 157–161.Google Scholar
7.Cozzens, J. H., Homological properties of the ring of differential polynomials, Bull. Amer. Math. Soc. 76 (1970), 75–79.CrossRefGoogle Scholar
8.Dischinger, M. F., Sur les anneaux fortement π-reguliers, C. R. Acad. Sci. Paris, Ser. A 283 (1976), 571–573.Google Scholar
9.Fisher, J. W., Von Neumann regular rings versus V-rings, Ring Theory, Marcel Dekker, 1974, 101–119.Google Scholar
10.Fisher, J. W. and Snider, R. L., On the von Neumann regularity of rings with regular prime factor rings, Pacific J. Math. 54 (1974), 135–144.CrossRefGoogle Scholar
11.Goodearl, K. R., Von Neumann Regular Rings (Pitman, 1979).Google Scholar
12.Hamsher, R. M., Commutative rings over which every module has a maximal submodule, Proc. Amer. Math. Soc. 18 (1967), 1133–1137.CrossRefGoogle Scholar
13.Kambara, H. and Oshiro, K., On P-exchange rings, Osaka J. Math. 25 (1988), 833–842.Google Scholar
14.Koifman, L. A., Rings over which every module has a maximal submodule, Math. Notes 7 (1970), 215–219.CrossRefGoogle Scholar
15.Lambek, J., Lectures on Rings and Modules (Chelsea, 1976).Google Scholar
16.Nicholson, W. K., Lifting idempotents and exchange rings, Trans. Amer. Math. Soc. 229 (1977), 269–278.CrossRefGoogle Scholar
17.Ramamurthi, V. S., Weakly regular rings, Canad. Math. Bull. 16 (1973), 317–321.CrossRefGoogle Scholar
18.Stenstrom, Bo, Rings of Quotients (Springer-Verlag, 1975).CrossRefGoogle Scholar
19.Stock, J., On rings whose projective modules have exchange property, J. Algebra 103 (1986), 437–453.CrossRefGoogle Scholar
20.Warfield, R. B., Exchange rings and decompositions of modules. Math. Ann. 199 (1972), 31–36.CrossRefGoogle Scholar
21.Xue, W., On weakly right duo rings, Riv. Mat. Univ. Parma 15 (1989), 211–217.Google Scholar
22.Yao, X., Weakly right duo rings, Pure Appl. Math. Sci. 21 (1985), 19–24.Google Scholar
23.Ming, R. Yue Chi, On annihilator ideals (IV), Riv. Mat. Univ. Parma 13 (1987), 19–27.Google Scholar