I have to find the projective dimension of Z/pZ as Z-module, Z/pZ-module and Z/(p^2)Z-module, p is prime.Could anyone help me?
let $\displaystyle M=\mathbb{Z}/p\mathbb{Z}.$ clearly $\displaystyle M$ as $\displaystyle \mathbb{Z}/p\mathbb{Z}$ module is free and so projective. thus $\displaystyle \text{proj} \cdot \dim_{\mathbb{Z}/p\mathbb{Z}} M = 0.$
$\displaystyle M$ as $\displaystyle \mathbb{Z}$ module is torsion and so it's not projective. now consider the exact sequence $\displaystyle 0 \to p\mathbb{Z} \to \mathbb{Z} \to M \to 0.$ as $\displaystyle \mathbb{Z}$ modules: $\displaystyle p\mathbb{Z} \cong \mathbb{Z}.$ hence $\displaystyle p\mathbb{Z}$ is projective. thus $\displaystyle \text{proj} \cdot \dim_{\mathbb{Z}} M = 1.$
finally looking at $\displaystyle M$ as $\displaystyle \mathbb{Z}/p^2 \mathbb{Z}$ module we have this exact sequence: $\displaystyle 0 \to p \mathbb{Z}/p^2 \mathbb{Z} \to \mathbb{Z}/p^2 \mathbb{Z} \to M \to 0.$ but as $\displaystyle \mathbb{Z}/p^2 \mathbb{Z}$ module: $\displaystyle p \mathbb{Z}/p^2 \mathbb{Z} \cong M.$ therefore: $\displaystyle \text{proj} \cdot \dim_{\mathbb{Z}/p^2 \mathbb{Z}} M = \infty.$