×

Generalized V-rings and von Neumann regular rings. (English) Zbl 0547.16006

Relations among the following conditions on a ring R are studied: right GV (all simple singular right R-modules are injective), FRI (all right ideals of R are idempotent), GFRI \((soc(R_ R)\) is projective and \(R/soc(R_ R)\) is FRI), right SI (all singular right R-modules are injective), ASR (R is regular and R/soc(R) is strongly regular), MERT (all maximal essential right ideals of R are two-sided). For instance, it is proved that R is right GV if and only if \(soc(R_ R)\) is projective and \(R/soc(R_ R)\) is a right V-ring. Assuming R is MERT, it is shown that R is ASR if and only if R is right GV and \(soc(R_ R)^ 2\cap J(R)=0,\) if and only if R is regular, if and only if R is FRI. Some earlier work on GV-rings may be found in J. S. Alin and E. P. Armendariz [Math. Scand. 23, 233-240 (1968; Zbl 0185.088)].
Reviewer: K.R.Goodearl

MSC:

16E50 von Neumann regular rings and generalizations (associative algebraic aspects)
16D50 Injective modules, self-injective associative rings

Citations:

Zbl 0185.088
PDFBibTeX XMLCite
Full Text: Numdam EuDML

References:

[1] G. Azumaya , Some properties of TTF-classes , Proc. of the Conference on orders, group rings and related topics, pp. 72 - 83 , Lecture Notes in Math. , vol. 353 , Springer-Verlag , 1973 . MR 338073 | Zbl 0272.16011 · Zbl 0272.16011
[2] G. Baccella , On flat factor rings and fully right idempotent rings , Ann. Univ. Ferrara , 26 ( 1980 ), pp. 125 - 141 . MR 608303 | Zbl 0456.16023 · Zbl 0456.16023
[3] G. Baccella , On C-semisimple rings. A study of the socle of a ring, Commun . in Algebra , 8 ( 10 ) ( 1980 ), pp. 889 - 909 . MR 573460 | Zbl 0434.16020 · Zbl 0434.16020
[4] G. Baccella : Weakly semiprime rings, Commun . in Algebra , 12 ( 4 ) ( 1984 ), 489 - 509 . MR 737259 | Zbl 0535.16004 · Zbl 0535.16004
[5] K. Byrd , Right self-injeotive rings whose essential right ideals are two-sided , Pacific J. Math. , 82 ( 1979 ), pp. 23 - 41 . Article | MR 549830 | Zbl 0416.16006 · Zbl 0416.16006
[6] J.H. Cozzens , Homological properties of the ring of differential polinomials , Bull. Amer. Math. Soc. , 76 ( 1970 ), pp. 75 - 79 . Article | MR 258886 | Zbl 0213.04501 · Zbl 0213.04501
[7] J.W. Fisher , Von Neumann regular rings versus V-rings, Proc. Univ. of Oklahoma ring theory symposium , Lect. Notes in Pure and Appl. Math. , vol. 7 , Marcel Dekker, Inc. , New York , 1974 . MR 332866 | Zbl 0256.16002 · Zbl 0256.16002
[8] K.R. Goodearl , Singular torsion and the splitting properties , Mem. Amer. Math. Soc. , No. 124 ( 1972 ). MR 340335 | Zbl 0242.16018 · Zbl 0242.16018
[9] K.R. Goodearl , Ring theory. Nonsingular rings and modules, Monographs and Textbooks in Math ., Marcel Dekker, Inc. , New York , 1976 . MR 429962 | Zbl 0336.16001 · Zbl 0336.16001
[10] K.R. Goodearl , Von Neumann regular rings, Monographs and Studies in Math ., Pitman Publ. Ltd , London , 1979 . MR 533669 | Zbl 0411.16007 · Zbl 0411.16007
[11] F. Hansen , Certain overrings of right hereditary, right Noetherian rings are V-rings , Proc. Amer. Math. Soc. , 52 ( 1975 ), pp. 85 - 90 . MR 376764 | Zbl 0311.16028 · Zbl 0311.16028
[12] J.P. Jans , Projective injective modules , Pacific J. Math. , 9 ( 1959 ), pp. 1103 - 1108 . Article | MR 112904 | Zbl 0231.16012 · Zbl 0231.16012
[13] G.O. Michler - O.E. Villamayor , On rings whose simple modules are injeotive , Journal of Algebra , 25 ( 1973 ), pp. 185 - 201 . MR 316505 | Zbl 0258.16023 · Zbl 0258.16023
[14] V.S. Ramamurthi - K.M. Rangaswamy , Generalized V-rings , Math. Scand. , 31 ( 1972 ), pp. 69 - 77 . MR 321974 | Zbl 0241.16008 · Zbl 0241.16008
[15] D.W. Sharpe - P. Vámos , Injective modules, Cambridge Tracts in Math., No. 62 , Cambridge University Press , Cambridge , 1972 . MR 360706 | Zbl 0245.13001 · Zbl 0245.13001
[16] R. Yue Chi Ming , On simple p-injeotive modules , Math. Japonicae , 19 ( 1974 ), pp. 173 - 176 . MR 399167 | Zbl 0263.16019 · Zbl 0263.16019
[17] R. Yue Chi Ming , On (Von Neumann) regular rings , Proc. Edinburg Math. Soc. , 19 (ser. II ), ( 1974-75 ), pp. 89 - 91 . MR 342553 | Zbl 0268.16007 · Zbl 0268.16007
[18] R. Yue Chi Ming , On Von Neumann Regular Rings, II , Math. Scand. , 39 ( 1976 ), pp. 167 - 170 . Article | MR 435131 | Zbl 0344.16011 · Zbl 0344.16011
[19] R. Yue Chi Ming , On V-rings and prime rings , Journal of Algebra , 62 ( 1980 ), pp. 13 - 20 . MR 561114 | Zbl 0429.16018 · Zbl 0429.16018
[20] R. Yue Chi Ming , On V-rings and unit-regular rings , Rend. Sem. Mat. Univ. Padova , 64 ( 1981 ), pp. 127 - 140 . Numdam | MR 636631 | Zbl 0474.16008 · Zbl 0474.16008
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.