| course |
course_year |
question_number |
tags |
title |
year |
Groups, Rings and Modules |
IB |
31 |
IB |
2019 |
Groups, Rings and Modules |
|
Paper 2, Section I, G |
2019 |
Let $R$ be an integral domain. A module $M$ over $R$ is torsion-free if, for any $r \in R$ and $m \in M, r m=0$ only if $r=0$ or $m=0$.
Let $M$ be a module over $R$. Prove that there is a quotient
$$q: M \rightarrow M_{0}$$
with $M_{0}$ torsion-free and with the following property: whenever $N$ is a torsion-free module and $f: M \rightarrow N$ is a homomorphism of modules, there is a homomorphism $f_{0}: M_{0} \rightarrow N$ such that $f=f_{0} \circ q$.