Previous |  Up |  Next


precobalanced sequence; cobalanced sequence; torsion-free image; pure submodule
The class of pure submodules ($\mathcal P$) and torsion-free images ($\mathcal R$) of finite direct sums of submodules of the quotient field of an integral domain were first investigated by M. C. R. Butler for the ring of integers (1965). In this case ${\mathcal P} = {\mathcal R}$ and short exact sequences of such modules are both prebalanced and precobalanced. This does not hold for integral domains in general. In this paper the notion of precobalanced sequences of modules is further investigated. It is shown that as in the case for abelian groups the exact sequence $ 0 \rightarrow M \rightarrow L \rightarrow T \rightarrow 0 $ with torsion $T$ is precobalanced precisely when it is cobalanced and in this case will split if $M$ is torsion-free of rank $1$. It is demonstrated that containment relationships between $\mathcal P$ and $\mathcal R$ for a domain $R$ are intimately related to the issue of when pure submodules of Butler modules are precobalanced. An analogous statement is made regarding the dual question of when torsion-free images of Butler modules are prebalanced.
[1] M. C. R. Butler: A class of torsion-free abelian groups of finite rank. Proc. London Math. Soc. 15 (1965), 680–698. MR 0218446 | Zbl 0131.02501
[2] M. C. R. Butler: Torsion-free modules and diagrams of vector spaces. Proc. Lond. Math. Soc., III. Ser. 18 (1968), 635–652. MR 0230767 | Zbl 0179.32603
[3] Laszlo Fuchs, Claudia Metelli: Countable Butler groups. Contemporary Mathematics 130 (1992), 133–143. DOI 10.1090/conm/130/1176115 | MR 1176115
[4] Laszlo Fuchs, Gert Viljoen: Note on extensions of Butler groups. Bull. Aust. Math. Soc. 41 (1990), 117–122. DOI 10.1017/S0004972700017901 | MR 1043972
[5] Anthony Giovannitti: A note on proper classes of exact sequences. Methods in Module Theory, Marcel Dekker, 1992, pp. 107–116. MR 1203802
[6] Anthony Giovannitti, Pat Goeters, Achim Kehrein: Prebalanced and balanced sequences of modules over domains. Commun. Algebra 31 (2003), 4817–4829. DOI 10.1081/AGB-120023134 | MR 1998030
[7] H. Pat Goeters: Butler modules over $1$-dimensional noetherian domains. Abelian Groups and Modules, Birkhäuser, Basel, 1999, pp. 149–165. MR 1735565
[8] H. Pat Goeters, Charles Vinsonhaler: Butler modules and Bext. Rings, Modules, Algebras, and Abelian Groups, Proceedings of the Algebra Conference—Venezia 2002, Marcel Dekker, New York, Lect. Notes Pure Appl. Math. 236 (2004), 307–319. MR 2050719
[9] Thomas W. Hungerford: Algebra. GTM 73, Springer, 1974. MR 0354211
[10] J. P. Jans: Rings and Homology. Holt, Rinehart and Winston, New York, 1964. MR 0163944 | Zbl 0141.02901
[11] Saunders MacLane: Homology. Springer, Berlin, 1963. MR 0156879
[12] Eben Matlis: One Dimensional Cohen-Macaulay Rings. Lect. Notes Math., Springer, Berlin, 1973. MR 0357391
[13] Eben Matlis: Torsion Free Modules. University of Chicago Press, Chicago, 1972. MR 0344237
[14] H. Matsumura: Commutative Ring Theory. Cambridge University Press, 1980. MR 0879273
[15] Fred Richman, Elbert A. Walker: Ext in preabelian categories. Pac. J. Math. 71 (1977), 521–535. DOI 10.2140/pjm.1977.71.521 | MR 0444742
[16] Joeseph Rotman: An Introduction to Homological Algebra. Academic Press, New York, 1979. MR 0538169
Partner of
EuDML logo