Let $R$ be a commutative ring (with 1) and $M$ a finitely generated $R$-module. We say that $M$ is rigid if for every finitely generated $R$-module $N$ whenever Tor$_i^R(M,N)=0$ then Tor$_j^R(M,N)=0$ for all $j\geq i$.
Could someone provide me with an example of a rigid $R$-module that is not a flat module or any other "nice" examples to keep in mind?
Thanks in advance.
Any finitely generated module over a regular local ring satisfies the Tor-rigid property, a result due to Lichtenbaum ("On the vanishing of Tor in regular local rings"). The proof occurs within the first five pages in the paper.
For a slightly less obvious but still perhaps too simple example, you could take the ring of power series $k[[x]]$ over a regular local ring $k$ and the module is $k$ where $x$ acts as zero; i.e. $k$ becomes a $k[[x]]$-module through the augmentation homomorphism $k[[x]]\to k$ given by $x\mapsto 0$.
For higher dimensional examples you could take similar examples over a power series ring in $n$ variables. If you want to come up with more examples yourself maybe you could try $k[[x,y]]$ and $k^n$ where $x$ and $y$ act as commuting nilpotent matrices, though I haven't verified that. I suggest playing with various regular local rings to learn more about this.