On quasi-duo rings

Yu, Hua-Ping

Glasgow mathematical journal, Tome 37 (1995) no. 1, pp. 21-31

Résumé

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.

Détail

DOI : 10.1017/S0017089500030342

Bibliographie

[1] 1.Bass, H., Finitistic dimension and a generalization of semiprimary rings, Trans. Amer. Math. Soc. 95 (1960), 466–488. Google Scholar | DOI

[2] 2.Brown, S. H., Rings over which every simple module is rationally complete, Canad. J. Math. 25 (1973), 693–701. Google Scholar | DOI

[3] 3.Burgess, W. D. and Stephenson, W., Rings all of whose Pierce stalks are local, Canad. Math. Bull. 22 (1979), 159–164. Google Scholar | DOI

[4] 4.Crawley, P. and Jonsson, B., Refinements for infinite direct decompositions of algebraic systems, Pacific J. Math. 14 (1964), 797–855. Google Scholar | DOI

[5] 5.Chandran, V. R., On a conjecture of Hyman Bass, Pure Appl. Math. Sci. 4 (1976), 125–131. Google Scholar

[6] 6.Courter, R. C., Finite dimensional right duo algebras are duo, Proc. Amer. Math. Soc. 84 (1982), 157–161. Google Scholar

[7] 7.Cozzens, J. H., Homological properties of the ring of differential polynomials, Bull. Amer. Math. Soc. 76 (1970), 75–79. Google Scholar | DOI

[8] 8.Dischinger, M. F., Sur les anneaux fortement π-reguliers, C. R. Acad. Sci. Paris, Ser. A 283 (1976), 571–573. Google Scholar

[9] 9.Fisher, J. W., Von Neumann regular rings versus V-rings, Ring Theory, Marcel Dekker, 1974, 101–119. Google Scholar

[10] 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. Google Scholar | DOI

[11] 11.Goodearl, K. R., Von Neumann Regular Rings (Pitman, 1979). Google Scholar

[12] 12.Hamsher, R. M., Commutative rings over which every module has a maximal submodule, Proc. Amer. Math. Soc. 18 (1967), 1133–1137. Google Scholar | DOI

[13] 13.Kambara, H. and Oshiro, K., On P-exchange rings, Osaka J. Math. 25 (1988), 833–842. Google Scholar

[14] 14.Koifman, L. A., Rings over which every module has a maximal submodule, Math. Notes 7 (1970), 215–219. Google Scholar | DOI

[15] 15.Lambek, J., Lectures on Rings and Modules (Chelsea, 1976). Google Scholar

[16] 16.Nicholson, W. K., Lifting idempotents and exchange rings, Trans. Amer. Math. Soc. 229 (1977), 269–278. Google Scholar | DOI

[17] 17.Ramamurthi, V. S., Weakly regular rings, Canad. Math. Bull. 16 (1973), 317–321. Google Scholar | DOI

[18] 18.Stenstrom, Bo, Rings of Quotients (Springer-Verlag, 1975). Google Scholar | DOI

[19] 19.Stock, J., On rings whose projective modules have exchange property, J. Algebra 103 (1986), 437–453. Google Scholar | DOI

[20] 20.Warfield, R. B., Exchange rings and decompositions of modules. Math. Ann. 199 (1972), 31–36. Google Scholar | DOI

[21] 21.Xue, W., On weakly right duo rings, Riv. Mat. Univ. Parma 15 (1989), 211–217. Google Scholar

[22] 22.Yao, X., Weakly right duo rings, Pure Appl. Math. Sci. 21 (1985), 19–24. Google Scholar

[23] 23.Ming, R. Yue Chi, On annihilator ideals (IV), Riv. Mat. Univ. Parma 13 (1987), 19–27. Google Scholar