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## Colloquium Mathematicum

1998 | 75 | 1 | 19-31
Tytuł artykułu

### Weak Baer modules over graded rings

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Słowa kluczowe
Czasopismo
Rocznik
Tom
Numer
Strony
19-31
Opis fizyczny
Daty
wydano
1998
otrzymano
1996-12-02
poprawiono
1997-03-03
Twórcy
autor
• Department of Mathematical Sciences, University of Wisconsin-Milwaukee, Milwaukee, Wisconsin 53201, U.S.A.
autor
• Department of Algebra and Analysis, University of Almería, 04071 Almería, Spain
Bibliografia
• [1] E. Cartan and S. Eilenberg, Homological Algebra, Princeton Univ. Press, 1956.
• [2] L. Fuchs and G. Viljoen, A weaker form of Baer's splitting problem over valuation domains, Quaestiones Math. 14 (1991), 227-236.
• [3] N C. Năstăsescu, Group rings of graded rings. Applications, J. Pure Appl. Algebra 33 (1984), 313-335.
• [4] C. Năstăsescu and B. Torrecillas, Relative graded Clifford theory, ibid. 83 (1992), 177-196.
• [5] C. Năstăsescu, M. Van den Bergh and F. Van Oystaeyen, Separable functors applied to graded rings, J. Algebra 123 (1989), 397-413.
• [6] C. Năstăsescu and F. Van Oystaeyen, Graded Ring Theory, North-Holland, Amsterdam, 1982.
• [7] P D. Passman, The Algebraic Structure of Group Rings, Wiley, New York, 1977.
• [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] B. D. Redman, and M. L. Teply, Flat Torsionfree Modules, in: Proc. 1993 Conf. on Commutative Algebra, Aguadulce, Spain, University of Almer\'\ia Press, 1995, 163-190.
• [10] S. Rim and M. Teply, Weak Baer modules localized with respect to a torsion theory, Czechoslovak Math. J., to appear.
• [11] R L. Rowen, Ring Theory, Academic Press, 1988.
• [12] S B. Stenström, Rings of Quotients, Springer, Berlin, 1975.
• [13] T M. Teply, Semicocritical Modules, Universidad de Murcia, 1987.
• [14] M. Teply and B. Torrecillas, A weaker form of Baer's splitting problem for torsion theories, Czechoslovak Math. J. 43 (1993), 663-674.
• [15] M. Teply and B. Torrecillas, Strongly graded rings with the Bounded Splitting Property, J. Algebra, to appear.
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