×

MC2 rings and WQD rings. (English) Zbl 1210.16004

The authors, in this paper, study left MC2 rings and left WQD rings. In Section 2, they prove that every left idempotent reflexive ring is a left MC2 ring. They prove that a left min-Abel ring \(R\) is left MC2 if and only if every simple projective left \(R\)-module is injective. They prove that every left WQD ring is directly finite. Finally, they get some equivalent conditions for left DS rings.
In Section 3, they study left SGP V-rings and strongly regular rings. They prove that a left SGP V-ring is left DS if and only if it is left MC2. They prove that a ring \(R\) is strongly regular if and only if it is left MC2, left WQD and left SGP V-ring. They generalize the result of Rege “a ring \(R\) is strongly regular if and only if it is left weakly regular and left quasi-duo” to left WQD rings.
Finally, in Section 4, they study rings admitting a maximal injective left ideal. They prove that a left MC2 ring with this property is left self-injective. This generalizes the result of Kim and Baik who proved that every left idempotent reflexive ring with an injective maximal left ideal is left self-injective.
They claim that their Corollary 4.6 generalizes the celebrated result of Osofsky “every left hereditary, left self-injective ring is semisimple Artinian”. This doesn’t seem to be correct as the hypothesis of Osofky’s result quoted above doesn’t seem to imply that the ring \(R\) is a left HI ring. Also, in the proofs of both (4) implies (5) of Theorem 2.3 and Theorem 4.1, the authors don’t seem to use the fact that \(L\) is an essential left ideal of \(R\).

MSC:

16D50 Injective modules, self-injective associative rings
16E50 von Neumann regular rings and generalizations (associative algebraic aspects)
16D25 Ideals in associative algebras
16D80 Other classes of modules and ideals in associative algebras
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] DOI: 10.1006/jabr.1996.6936 · Zbl 0884.16002
[2] DOI: 10.1017/S0017089500030342 · Zbl 0819.16001
[3] Wei, Southeast Asian Bull. Math. 31 pp 1009– (2007)
[4] DOI: 10.1007/s10114-004-0416-1 · Zbl 1086.16004
[5] Song, Kyungpook Math. J. 45 pp 357– (2005)
[6] DOI: 10.2307/2047200 · Zbl 0657.16015
[7] Rege, Math Japonica 31 pp 927– (1986)
[8] Ming, Glasnik Mat. 18 pp 25– (1983)
[9] Osofsky, Pacific J. Math. 14 pp 645– (1964) · Zbl 0145.26601
[10] DOI: 10.1080/00927878108822678 · Zbl 0468.16024
[11] DOI: 10.1081/AGB-100002399 · Zbl 0983.16002
[12] DOI: 10.1016/j.jpaa.2004.08.011 · Zbl 1071.16003
[13] DOI: 10.1006/jabr.1996.6796 · Zbl 0879.16002
[14] DOI: 10.1080/00927879908826551 · Zbl 0923.16008
[15] DOI: 10.1006/jabr.1995.1117 · Zbl 0839.16004
[16] Kim, Kyungpook Math. J. 46 pp 597– (2006)
[17] Kim, Proc. Japan. Acad. 81 pp 125– (2005)
[18] Kaplansky, Rings of operators (1968) · Zbl 0174.18503
[19] DOI: 10.1155/2007/63171 · Zbl 1152.16009
[20] DOI: 10.1080/00927879308824811 · Zbl 0803.16010
[21] Wei, Int. Electron. J. Algebra 2 pp 1– (2007)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.