Weak Baer modules over graded rings

Mark Teply; Blas Torrecillas

Colloquium Mathematicae (1998)

  • Volume: 75, Issue: 1, page 19-31
  • ISSN: 0010-1354

Abstract

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In [2], Fuchs and Viljoen introduced and classified the B * -modules for a valuation ring R: an R-module M is a B * -module if E x t R 1 ( M , X ) = 0 for each divisible module X and each torsion module X with bounded order. The concept of a B * -module was extended to the setting of a torsion theory over an associative ring in [14]. In the present paper, we use categorical methods to investigate the B * -modules for a group graded ring. Our most complete result (Theorem 4.10) characterizes B * -modules for a strongly graded ring R over a finite group G with | G | − 1 ∈ R . Motivated by the results of [8], [9], [10] and [15], we also study the condition that every non-singular R-module is a B ∗ -module with respect to the Goldie torsion theory; for the case in which R is a strongly graded ring over a group, extensive information is obtained for group rings of abelian, solvable and polycyclic-by-finite groups.

How to cite

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Teply, Mark, and Torrecillas, Blas. "Weak Baer modules over graded rings." Colloquium Mathematicae 75.1 (1998): 19-31. <http://eudml.org/doc/210525>.

@article{Teply1998,
abstract = {In [2], Fuchs and Viljoen introduced and classified the $B^*$-modules for a valuation ring R: an R-module M is a $B^*$-module if $Ext^1_R(M,X)=0$ for each divisible module X and each torsion module X with bounded order. The concept of a $B^*$-module was extended to the setting of a torsion theory over an associative ring in [14]. In the present paper, we use categorical methods to investigate the $B^*$-modules for a group graded ring. Our most complete result (Theorem 4.10) characterizes $B^*$-modules for a strongly graded ring R over a finite group G with $|G|^\{−1\} \in R$. Motivated by the results of [8], [9], [10] and [15], we also study the condition that every non-singular R-module is a $B^∗$-module with respect to the Goldie torsion theory; for the case in which R is a strongly graded ring over a group, extensive information is obtained for group rings of abelian, solvable and polycyclic-by-finite groups.},
author = {Teply, Mark, Torrecillas, Blas},
journal = {Colloquium Mathematicae},
keywords = {-modules; strongly graded rings; graded modules; hereditary torsion theories; group rings; separable functors; bounded splitting property; Grothendieck categories; categories of modules; graded torsion theories},
language = {eng},
number = {1},
pages = {19-31},
title = {Weak Baer modules over graded rings},
url = {http://eudml.org/doc/210525},
volume = {75},
year = {1998},
}

TY - JOUR
AU - Teply, Mark
AU - Torrecillas, Blas
TI - Weak Baer modules over graded rings
JO - Colloquium Mathematicae
PY - 1998
VL - 75
IS - 1
SP - 19
EP - 31
AB - In [2], Fuchs and Viljoen introduced and classified the $B^*$-modules for a valuation ring R: an R-module M is a $B^*$-module if $Ext^1_R(M,X)=0$ for each divisible module X and each torsion module X with bounded order. The concept of a $B^*$-module was extended to the setting of a torsion theory over an associative ring in [14]. In the present paper, we use categorical methods to investigate the $B^*$-modules for a group graded ring. Our most complete result (Theorem 4.10) characterizes $B^*$-modules for a strongly graded ring R over a finite group G with $|G|^{−1} \in R$. Motivated by the results of [8], [9], [10] and [15], we also study the condition that every non-singular R-module is a $B^∗$-module with respect to the Goldie torsion theory; for the case in which R is a strongly graded ring over a group, extensive information is obtained for group rings of abelian, solvable and polycyclic-by-finite groups.
LA - eng
KW - -modules; strongly graded rings; graded modules; hereditary torsion theories; group rings; separable functors; bounded splitting property; Grothendieck categories; categories of modules; graded torsion theories
UR - http://eudml.org/doc/210525
ER -

References

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  1. [1] E. Cartan and S. Eilenberg, Homological Algebra, Princeton Univ. Press, 1956. 
  2. [2] L. Fuchs and G. Viljoen, A weaker form of Baer's splitting problem over valuation domains, Quaestiones Math. 14 (1991), 227-236. Zbl0738.13017
  3. [3] N C. Năstăsescu, Group rings of graded rings. Applications, J. Pure Appl. Algebra 33 (1984), 313-335. Zbl0542.16003
  4. [4] C. Năstăsescu and B. Torrecillas, Relative graded Clifford theory, ibid. 83 (1992), 177-196. 
  5. [5] C. Năstăsescu, M. Van den Bergh and F. Van Oystaeyen, Separable functors applied to graded rings, J. Algebra 123 (1989), 397-413. 
  6. [6] C. Năstăsescu and F. Van Oystaeyen, Graded Ring Theory, North-Holland, Amsterdam, 1982. 
  7. [7] P D. Passman, The Algebraic Structure of Group Rings, Wiley, New York, 1977. Zbl0368.16003
  8. [8] B. D. Redman, Jr., and M. L. Teply, Torsionfree B^*-modules, in: Ring Theory, Proc. 21st Ohio State/Denison Conf., Granville, Ohio, 1992, World Scientific, River Edge, N.J., 1993, 314-328. 
  9. [9] B. D. Redman, and M. L. Teply, Flat Torsionfree Modules, in: Proc. 1993 Conf. on Commutative Algebra, Aguadulce, Spain, University of AlmerPress, 1995, 163-190. 
  10. [10] S. Rim and M. Teply, Weak Baer modules localized with respect to a torsion theory, Czechoslovak Math. J., to appear. Zbl0931.16014
  11. [11] R L. Rowen, Ring Theory, Academic Press, 1988. 
  12. [12] S B. Stenström, Rings of Quotients, Springer, Berlin, 1975. Zbl0296.16001
  13. [13] T M. Teply, Semicocritical Modules, Universidad de Murcia, 1987. 
  14. [14] M. Teply and B. Torrecillas, A weaker form of Baer's splitting problem for torsion theories, Czechoslovak Math. J. 43 (1993), 663-674. Zbl0808.16033
  15. [15] M. Teply and B. Torrecillas, Strongly graded rings with the Bounded Splitting Property, J. Algebra, to appear. 

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