 This lecture is part of an online commutative algebra course and will be the last of a sequence of about eight lectures on the different types of modules that are. And this one will be about torsion free modules. It'll be quite short because frankly, there's not a lot to say about torsion free modules. So I'll just summarize what we've done so far. So we've defined these different sorts of modules free, which implies stably free, which implies locally free, which implies projective, which implies stock wise free, which implies flat, which is the really important one. And this implies torsion free, and this implies that it's a co-primary, at least over integral domains. And what we're going to do this lecture is just quickly summarize the properties of the last two in this list. So first of all, we should define what a torsion free module is over an integral domain. We say M is torsion free if XM equals naught implies X equals naught or M equals naught here. X is in the ring R and M is in the module M, of course. So over a general ring, there are actually several slightly different definitions of torsion free in the literature. So one of them might be for a general ring, if XM equals naught where X is in R and M is in R. This implies M equals naught or X is a zero divisor. The reason for this is that if X has zero divisors, then you discover that with this definition almost nothing is torsion free. So you don't often use torsion free modules for general rings, so you don't need to pay too much attention to this. In fact, frankly speaking, you don't often use torsion free modules even for integral domains. So now we should check that flat implies torsion free. And this is quite easy because if X is not a zero divisor, this implies the sequence naught goes to R, goes to R, goes to R over XR goes to naught where this is multiplication by X is exact. And if M is flat, we can tensor this and still get an exact sequence. So we get naught goes to M, goes to XM is exact. We don't worry about the right hand part of this. So X is not a zero divisor on the module M, so M is torsion free. On the other hand, there are many examples of torsion free modules that are not flat. Well, first of all, over principle ideal domains we can't do this. So let's just remark that over the principle ideal domain torsion free is the same as flat. And this is probably the only reason why people use torsion free modules at all. You quite often use them over principle ideal domains and there they have the advantage that torsion free is a little bit easier to define than flatness. But it turns out that you may as well just forget about torsion free-ness and use flatness because flatness works nicely over other rings. The fact that over principle ideal domain to torsion free modules are flat follows from the fact that M is flat if and only if the M tends to with I is a submodule of M for all ideals. I, this works for any ring, not necessarily a principle ideal domain. And if it's a principle ideal domain then any ideal is principle. So torsion free-ness implies this property. Actually, this is also true for data kind domains. Torsion free also implies flatness. Anyway, got a bit sidetracked. We're really trying to find an example of a module that's torsion free but not flat. And the simplest example of this is you take the ideal to be generated by X and Y in the ring of polynomials over X and Y and then I is torsion free but not flat. So it's obviously torsion free. To see it's not flat you can either cheat and note that tor one of I with K is not zero. And this is cheating a bit because we haven't actually defined tor groups or calculated them but in a few lectures time we'll be checking this. Or you can just check directly that if you take the exact sequence nought goes to I, goes to K, X, Y and you tense it with I. This is I tense I goes to I. This is not exact but to do this you have to calculate I tense I which I'm feeling too lazy to do just a moment. So well last time we showed that over local rings things like flatness and projectiveness and so on were all the same. So you could ask is a torsion free module over a flat ring. So is a torsion free module over a local ring also flat? And the answer is no. We can take an example with our local of a ring that's of a module that's torsion free but not flat. And here we just sort of do almost exactly the same as this we just take R to be the ring of formal power series over a field instead of the polynomial ring. And again, we take I to be X, Y and then the ideal is torsion free but not flat. So torsion free doesn't behave all that nicely even for local rings. I'll just finish off by commenting on the relation between torsion free and co-primary. At least over integral domains. So you recall that co-primary means there's only one associated prime and an associated prime is a prime that's the annihilator of some element M in the module. And if the module is torsion free then the annihilator of any non-zero element is always just zero. So the only associated prime is zero. So torsion free implies co-primary over integral domains. Co-primary does not imply torsion free and this is very easy. We can just take the module M to be Z modulo two Z and the ring R to be the integer Z. And this obviously isn't torsion free and it obviously is co-primary because the only associated prime is two. Okay, so that's enough about torsion free modules. So far in these lectures I've been regularly postponing proving various things on grounds that be a lot easier once we've done homological algebra and defined torsion groups. So next lecture I'm going to start several lectures on an introduction to homological algebra where we will fill in all these missing details I've been postponing.