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On simple singular GP-injective modules. (English) Zbl 0923.16008

A right \(R\)-module \(M\) is called generalized right principally injective (simply right GP-injective) if, for any \(0\neq a\in R\), there exists a positive integer \(n=n(a)\) depending on \(a\) such that \(a^n\neq 0\) and any right \(R\)-homomorphism from \(a^nR\) to \(M\) extends to one of \(R\) to \(M\).
Von Neumann regularity of rings whose simple singular \(R\)-modules are GP-injective is investigated. It is proved that a ring \(R\) is strongly regular if and only if \(R\) is a weakly right duo ring whose simple singular right \(R\)-modules are GP-injective. Also it is shown that a ring \(R\) is either a strongly right bounded ring or a zero insertive ring in which simple singular right \(R\)-modules are GP-injective is reduced weakly regular.
Reviewer: J.K.Park (Pusan)

MSC:

16E50 von Neumann regular rings and generalizations (associative algebraic aspects)
16D50 Injective modules, self-injective associative rings
16D60 Simple and semisimple modules, primitive rings and ideals in associative algebras
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References:

[1] Alin J.S., Math. Scand 23 pp 233– (1968)
[2] Baccella G., Rend. Sem. Mat. Univ. Padova 72 pp 117– (1984)
[3] Birkenmeier G.F., Proc. Biennial Ohio State-Denison Conference pp 102–
[4] DOI: 10.1090/S0002-9939-1994-1231028-7
[5] DOI: 10.1016/S0022-4049(96)00011-4 · Zbl 0870.16006
[6] Brown S.H., Canad. J. Math pp 693– (1973) · Zbl 0229.16017
[7] DOI: 10.1080/00927879408825068 · Zbl 0810.16011
[8] Ding N., Math. Japonica 40 pp 191– (1994)
[9] Faith C., Lonodn Math. Soc. Lecture Notes Series 88 (1984)
[10] Gupta V., Math. J. Okayama Univ 19 pp 123– (1977)
[11] Habeb J.M., Math. J. Okayama Univ 32 pp 73– (1990)
[12] Hirano Y., Math. J. Okayama Univ 20 pp 141– (1978)
[13] Hirano Y., Hiroshima Math. J 9 pp 137– (1979)
[14] Ming R.Y.C., Math. Scand 39 pp 167– (1976)
[15] Ming R.Y.C., J. Algebra 62 pp 13– (1980) · Zbl 0429.16018
[16] Ming R.Y.C., Riv. Math. Univ. Parma 11 pp 101– (1985)
[17] Ming R.Y.C., Riv. Math. Univ. Parma 13 pp 19– (1987)
[18] Ming R.Y.C., Portugaliae Math 44 pp 101– (1987)
[19] DOI: 10.1080/00927879508825543 · Zbl 0840.16006
[20] DOI: 10.4153/CMB-1973-051-7 · Zbl 0241.16007
[21] Ramamurthi V.S., Math. Scand 31 pp 69– (1972)
[22] Rege M.B., Math. Japon-ica 31 pp 927– (1986)
[23] Shin G., Trans. Amer. Math. Soc 84 pp 43– (1973)
[24] Tominaga H., Math. J. Okayama Univ 84 pp 171– (1977)
[25] Yao Xue, Pure and Applied Math. Sciences 21 pp 19– (1985)
[26] DOI: 10.1017/S0017089500030342 · Zbl 0819.16001
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