On a generalization of Q I -rings

Josef Jirásko

Commentationes Mathematicae Universitatis Carolinae (1999)

  • Volume: 40, Issue: 3, page 441-446
  • ISSN: 0010-2628

Abstract

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In this paper rings for which every s -torsion quasi-injective module is weakly s -divisible for a hereditary preradical s are characterized in terms of the properties of the corresponding lattice of the (hereditary) preradicals. In case of a stable torsion theory these rings coincide with T Q I -rings investigated by J. Ahsan and E. Enochs in [1]. Our aim was to generalize some results concerning Q I -rings obtained by J.S. Golan and S.R. L’opez-Permouth in [12]. A characterization of the Q I -property in the category σ [ M ] then follows as a consequence.

How to cite

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Jirásko, Josef. "On a generalization of $QI$-rings." Commentationes Mathematicae Universitatis Carolinae 40.3 (1999): 441-446. <http://eudml.org/doc/248411>.

@article{Jirásko1999,
abstract = {In this paper rings for which every $s$-torsion quasi-injective module is weakly $s$-divisible for a hereditary preradical $s$ are characterized in terms of the properties of the corresponding lattice of the (hereditary) preradicals. In case of a stable torsion theory these rings coincide with $TQI$-rings investigated by J. Ahsan and E. Enochs in [1]. Our aim was to generalize some results concerning $QI$-rings obtained by J.S. Golan and S.R. L’opez-Permouth in [12]. A characterization of the $QI$-property in the category $\sigma [M]$ then follows as a consequence.},
author = {Jirásko, Josef},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {$s$-$QI$-rings; $s$-stable preradicals; weakly $s$-divisible modules; $s$-tight modules; QI-rings; -stable preradicals; weakly divisible modules; tight modules; hereditary preradicals; quasi-injective modules; injective hulls},
language = {eng},
number = {3},
pages = {441-446},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On a generalization of $QI$-rings},
url = {http://eudml.org/doc/248411},
volume = {40},
year = {1999},
}

TY - JOUR
AU - Jirásko, Josef
TI - On a generalization of $QI$-rings
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1999
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 40
IS - 3
SP - 441
EP - 446
AB - In this paper rings for which every $s$-torsion quasi-injective module is weakly $s$-divisible for a hereditary preradical $s$ are characterized in terms of the properties of the corresponding lattice of the (hereditary) preradicals. In case of a stable torsion theory these rings coincide with $TQI$-rings investigated by J. Ahsan and E. Enochs in [1]. Our aim was to generalize some results concerning $QI$-rings obtained by J.S. Golan and S.R. L’opez-Permouth in [12]. A characterization of the $QI$-property in the category $\sigma [M]$ then follows as a consequence.
LA - eng
KW - $s$-$QI$-rings; $s$-stable preradicals; weakly $s$-divisible modules; $s$-tight modules; QI-rings; -stable preradicals; weakly divisible modules; tight modules; hereditary preradicals; quasi-injective modules; injective hulls
UR - http://eudml.org/doc/248411
ER -

References

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  2. Al-Huzali, Jain S.K., López-Permouth S.R., On the weak relative injectivity of rings and modules, Lecture Notes in Math. 1448, Springer Verlag, 1990, pp.93-98. MR1084625
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  9. Byrd K.A., Rings, whose quasi-injective modules are injective, Proc. Amer. Math. Soc. 33 (1972), 235-240. (1972) Zbl0214.05605MR0310009
  10. Gabriel P., Des catégories abéliennes, Bull. Soc. Math. France 90 (1962), 323-448. (1962) Zbl0201.35602MR0232821
  11. Golan J.S., Torsion Theories, Longman Scientific and Technical, Harlow, 1986. Zbl0695.16021MR0880019
  12. Golan J.S., López-Permouth S.R., QI-filters and tight modules, Comm. Algebra 19 (8) (1991), 2217-2229. (1991) MR1123120
  13. Jain S.K., López-Permouth S.R., Singh S., On a class of QI-rings, Glasgow Math. J. 34 (1992), 75-81. (1992) MR1145633
  14. Jirásko J., Generalized injectivity, Comment. Math. Univ. Carolinae 16 (1975), 621-636. (1975) MR0399165
  15. Morimoto S., Weakly divisible and divisible modules, Tsukuba J. Math. 6 (2) (1982), 195-200. (1982) Zbl0529.16019MR0705113

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