| course |
course_year |
question_number |
tags |
title |
year |
Groups, Rings and Modules |
IB |
33 |
IB |
2013 |
Groups, Rings and Modules |
|
Paper 4, Section II, 11G |
2013 |
Let $R$ be an integral domain, and $M$ be a finitely generated $R$-module.
(i) Let $S$ be a finite subset of $M$ which generates $M$ as an $R$-module. Let $T$ be a maximal linearly independent subset of $S$, and let $N$ be the $R$-submodule of $M$ generated by $T$. Show that there exists a non-zero $r \in R$ such that $r x \in N$ for every $x \in M$.
(ii) Now assume $M$ is torsion-free, i.e. $r x=0$ for $r \in R$ and $x \in M$ implies $r=0$ or $x=0$. By considering the map $M \rightarrow N$ mapping $x$ to $r x$ for $r$ as in (i), show that every torsion-free finitely generated $R$-module is isomorphic to an $R$-submodule of a finitely generated free $R$-module.