/sci/ - Science & Math

>>11697013
Imagine showing that modules over a certain kind of ring have a property, pick a ring you want to study and construct a counter-example. I'm feeling silly, so I give you a very very very stupid example: the ring of integers is not a field, because [math]\text{Tor}(\mathbb{Z}, \mathbb{Z}/2\mathbb{Z}) \neq 0[/math]. You can also weaken isomorphism to Morita equivalence which is a property of rings but an equivalence between the categories of modules. This gives you more wiggle room, just like weakening homeomorphism to homotopy equivalence does in topology.