On quasi-duo rings. (English) Zbl 0819.16001

Yu, Hua-Ping

Glasg. Math. J. 37, No. 1, 21-31 (1995).

Recall that a ring \(R\) (with identity) is left quasi-duo if every maximal left ideal of \(R\) is two-sided. The author proves that the following are equivalent for a left quasi-duo ring \(R\): (1) every projective left \(R\)- module has the exchange property: (2) the free left \(R\)-module \(R^{(N)}\) has the finite exchange property; (3) \(R/J(R)\) is von Neumann regular and \(J(R)\) is left \(T\)-nilpotent; (4) every non-zero left \(R\)-module has a maximal submodule. In this event, the author also shows that \(R\) is left perfect if \(R\) contains no infinite set of orthogonal idempotents. This latter result generalizes a theorem of the reviewer [Riv. Mat. Univ. Parma, IV. Ser. 15, 211-217 (1989; Zbl 0763.16001)].

Reviewer: Xue Weimin (Fouzhou)

MSC:

16D25	Ideals in associative algebras
16D70	Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras)
16U80	Generalizations of commutativity (associative rings and algebras)
16E50	von Neumann regular rings and generalizations (associative algebraic aspects)
16L30	Noncommutative local and semilocal rings, perfect rings

Keywords:

maximal left ideals; left quasi-duo rings; projective left modules; finite exchange property; von Neumann regular rings; left \(T\)-nilpotent rings; maximal submodules; left perfect rings; orthogonal idempotents

Citations:

Zbl 0763.16001

References:

[1]	Yao, Pure Appl. Math. Sci. 21 pp 19– (1985)
[2]	DOI: 10.2307/1998510 · Zbl 0352.16006
[3]	DOI: 10.1007/BF01419573 · Zbl 0228.16012
[4]	Dischinger, C. R. Acad. Sci. Paris, Ser. A 283 pp 571– (1976)
[5]	DOI: 10.1090/S0002-9904-1970-12370-9 · Zbl 0213.04501
[6]	DOI: 10.2307/2043655 · Zbl 0495.16013
[7]	Chandran, Pure Appl. Math. Sci. 4 pp 125– (1976)
[8]	Crawley, Pacific J. Math. 14 pp 797– (1964) · Zbl 0134.25504
[9]	Burgess, Canad. Math. Bull. 22 pp 159– (1979) · Zbl 0411.16009
[10]	Brown, Canad. J. Math. 25 pp 693– (1973) · Zbl 0229.16017
[11]	DOI: 10.2307/1993568 · Zbl 0094.02201
[12]	DOI: 10.1016/0021-8693(86)90145-6 · Zbl 0603.16016
[13]	Stenstrom, Rings of Quotients (1975)
[14]	Ramamurthi, Canad. Math. Bull. 16 pp 317– (1973) · Zbl 0241.16007
[15]	Lambek, Lectures on Rings and Modules (1976)
[16]	Koifman, Math. Notes 7 pp 215– (1970) · Zbl 0202.32702
[17]	Kambara, Osaka J. Math. 25 pp 833– (1988)
[18]	DOI: 10.2307/2035815 · Zbl 0156.04303
[19]	Goodearl, Von Neumann Regular Rings (1979)
[20]	Fisher, Pacific J. Math. 54 pp 135– (1974) · Zbl 0301.16015
[21]	Ming, Riv. Mat. Univ. Parma 13 pp 19– (1987)
[22]	Xue, Riv. Mat. Univ. Parma 15 pp 211– (1989)