# proof of finitely generated torsion-free modules over Prüfer domains

Let $M$ be a finitely generated^{} torsion-free module over a Prüfer domain $R$ with field of fractions^{} $k$. We show that $M$ is isomorphic^{} to a direct sum^{} (http://planetmath.org/DirectSum) of finitely generated ideals in $R$.

We shall write $k\otimes M$ for the vector space^{} over $k$ generated by $M$. This is just the localization^{} (http://planetmath.org/LocalizationOfAModule) of $M$ at $R\setminus \{0\}$ and, as $M$ is torsion-free, the natural map $M\to k\otimes M$ is one-to-one and we can regard $M$ as a subset of $k\otimes M$.

As $M$ is finitely generated, the vector space $k\otimes M$ will finite dimensional (http://planetmath.org/Dimension2), and we use induction^{} on its dimension^{} $n$.
Supposing that $n>0$, choose any basis ${e}_{1},\mathrm{\dots},{e}_{n}$ and define the linear map $f:k\otimes M\to k$ by projection^{} (http://planetmath.org/Projection) onto the first component^{},

$$f({x}_{1}{e}_{1}+\mathrm{\cdots}+{x}_{n}{e}_{n})={x}_{1}.$$ |

Restricting to $M$, this gives a nonzero map $M\to k$. Furthermore, as $M$ is finitely generated, $f(M)$ will be a finitely generated fractional ideal^{} in $k$. Choosing any nonzero $c\in R$ such that $\U0001d51e\equiv cf(M)\subseteq R$,

$$g:M\to \U0001d51e,g(u)=cf(u)$$ |

defines a homorphism from $M$ onto the nonzero and finitely generated ideal $\U0001d51e$. As $R$ is Prüfer and invertible ideals are projective, $g$ has a right-inverse $h:\U0001d51e\to M$.
Then $h$ has the left-inverse $g$ and is one-to-one, so defines an isomorphism^{} between $\U0001d51e$ and its image (http://planetmath.org/ImageOfALinearTransformation). We decompose $M$ as the direct sum of the kernel of $g$ and the image of $h$,

$$M=\mathrm{ker}(g)\oplus \mathrm{Im}(h)\cong \mathrm{ker}(g)\oplus \U0001d51e.$$ |

Projection from the finitely generated module $M$ onto $\mathrm{ker}(g)$ shows that it is finitely generated and,

$$\mathrm{dim}(k\otimes \mathrm{ker}(g))=\mathrm{dim}(k\otimes M)-\mathrm{dim}(k\otimes \U0001d51e)=n-1.$$ |

So, the result follows from applying the induction hypothesis to $\mathrm{ker}(g)$.

Title | proof of finitely generated torsion-free modules over Prüfer domains |
---|---|

Canonical name | ProofOfFinitelyGeneratedTorsionfreeModulesOverPruferDomains |

Date of creation | 2013-03-22 18:36:14 |

Last modified on | 2013-03-22 18:36:14 |

Owner | gel (22282) |

Last modified by | gel (22282) |

Numerical id | 4 |

Author | gel (22282) |

Entry type | Proof |

Classification | msc 13F05 |

Classification | msc 13C10 |