# Torsion-free modules over a general ring

I want to know how to prove that a torsion-free module over a general ring is flat. In Lectures on Rings and Modules, T.Y. Lam proves this in the case where your ring is an integral domain. Can you please help me prove it or recommend some books or articles concerning this? Thanks!

• Even over a domain a torsion free module is certainly not flat in general. – Georges Elencwajg Sep 6 '11 at 14:33
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• The integral domains that satisfy this property (every torsionfree module is flat) are exactly the so called Prüfer domains. – Johannes Hahn Sep 6 '11 at 21:33
• I'm not sure there's a consensus for a definition of torsion-free module over a ring which is not a domain. Please include a definition, or restrict to domains. – YCor Sep 23 '16 at 8:59

## 2 Answers

The best book for such questions in my opinion is the one you're already reading: "Lectures on Modules and Rings" by Lam. Indeed, on page 127 he provides a counter-example to your claim that torsion-free implies flat. Probably you meant the converse, which does hold: Any flat module is torsion-free. This is also on page 127.

Here's Lam's counter-example...Let $R=k[x,y]$ where $k$ is any commutative domain. Then $M=(x,y)$ is torsion-free because there are no relations on $x$ or $y$. However, $M$ is not flat. To see this set $S=R/(x)\cong k[y]$ so that $M\otimes_R S = M\otimes_R R/(x) \cong M/xM \cong (x,y)/(x^2,yx)$. If $M$ is flat over $R$ then $M\otimes_R S$ is flat over $S$ and hence torsion-free. This is a contradiction because $yx=0$ but $y\neq 0$.

• Here is a geometric proof of non-flatness of the ideal $M\subset R$.If $M$ were flat, it would be projective (flat+finitely presented $\Rightarrow$ projective), necessarily invertible since it is an ideal. But an invertible ideal has height one: contradiction, since $M$ trivially has height $\geq 2$ [actually $2$, of course]. A variant: a rank one projective module on $\mathbb A_k^2$ is trivial, so $M$ would be a free ideal, which means a principal ideal: this is obviously false. – Georges Elencwajg Sep 6 '11 at 19:02
• Thanks, Georges, that is very nice. I forgot about the connection between invertible ideals and projective ideals, but with it I think your proof is much more elegant – David White Sep 6 '11 at 19:36
• I, on the other hand, like the directness of your astute base change to $S$, David! And, also, Einstein remarked "If you are out to describe the truth, leave elegance to the tailor. " thinkexist.com/quotation/… – Georges Elencwajg Sep 6 '11 at 20:19

thank for your all answers! Here is my ideas: How can i prove the following proposition: " If every finitely generated ideal of R is principal, then a torsion - free R-module is flat"

Because most of books i have prove this property when R is integral domain, while i want to know how can we prove when R is general ring.

• You can prove the proposition in the usual way. Let $M$ be your module. You need to show that the canonical map $f: I\otimes M\to IM$ is an isomorphism for any ideal $I$ of $R$ of finite type. Write $I=aR$ and consider the map $M\to I\otimes M$, $x\mapsto a\otimes x$ and $h=fg : x\to ax$. Then $f,g$ are surjective by construction, and $h$ is an isomorphism because $M$ is torsion free. So $f$ is an isomorphism. – Qing Liu Sep 10 '11 at 17:38
• Please do not use answers to do anything except answering questions. – Mariano Suárez-Álvarez Oct 19 '13 at 19:00