Que $\newcommand\ZZ{\mathbb Z}R=\ZZ/p^n$, $n>1$, ser su anillo y considerar el módulo $M=\ZZ/p$. El mapa obvia $R\to M$ tiene kernel el ideal generado por $p$, por lo que la secuencia $R\xrightarrow{p}R\to M$ es exacta. Se genera el núcleo del mapa $p:R\to R$ $p^{n-1}\in R$. Así tenemos una secuencia exacta de la forma $$R\xrightarrow{\quad p^{n-1}\quad} R\xrightarrow{\quad p\quad}R\to M$$ Now the kernel of $p^{n-1}:R\to R $ is precisely the ideal generated by $p $, so we can extend the exact sequence to $$R\xrightarrow{\quad p\quad}R\xrightarrow{\quad p^{n-1}\quad} R\xrightarrow{\quad p\quad}R\to M$$ We can continue in this way, and conclude that there is a projective resolution of $M $ as an $R $-module which is of infinite length and with maps $p $ and $p ^ {n-1} $, repeaing with period $2$: $$\cdots \to R\xrightarrow{\quad p\quad}R\xrightarrow{\quad p^{n-1}\quad} R\xrightarrow{\quad p\quad}R\xrightarrow{\quad p^{n-1}\quad} R\xrightarrow{\quad p\quad}R\to M$$ To compute $ Tor $, we drop the $M $ on the right, apply the functor $ (-) \otimes_R M $ and compute the homology of the resulting complex. If you do that, you will find that the differentials in the complex are all zero, and all the modules isomorphic to $% M $: it follows that $\cong M$ for all $i\geq0$ Tor_i (M, M).
