Precobalanced and cobalanced sequences of modules over domains

Anthony Giovannitti; H. Pat Goeters

Mathematica Bohemica (2007)

  • Volume: 132, Issue: 1, page 35-42
  • ISSN: 0862-7959

Abstract

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The class of pure submodules ( 𝒫 ) and torsion-free images ( 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 𝒫 = 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 → M → L → T → 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 𝒫 and 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.

How to cite

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Giovannitti, Anthony, and Goeters, H. Pat. "Precobalanced and cobalanced sequences of modules over domains." Mathematica Bohemica 132.1 (2007): 35-42. <http://eudml.org/doc/250252>.

@article{Giovannitti2007,
abstract = {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.},
author = {Giovannitti, Anthony, Goeters, H. Pat},
journal = {Mathematica Bohemica},
keywords = {precobalanced sequence; cobalanced sequence; torsion-free image; pure submodule; torsion-free image; pure submodule},
language = {eng},
number = {1},
pages = {35-42},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Precobalanced and cobalanced sequences of modules over domains},
url = {http://eudml.org/doc/250252},
volume = {132},
year = {2007},
}

TY - JOUR
AU - Giovannitti, Anthony
AU - Goeters, H. Pat
TI - Precobalanced and cobalanced sequences of modules over domains
JO - Mathematica Bohemica
PY - 2007
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 132
IS - 1
SP - 35
EP - 42
AB - 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.
LA - eng
KW - precobalanced sequence; cobalanced sequence; torsion-free image; pure submodule; torsion-free image; pure submodule
UR - http://eudml.org/doc/250252
ER -

References

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