Bockstein on a cyclic Moore space ($\beta :H^i(M(\mathbb{Z}/n,i);\mathbb{Z}/n)\stackrel{\sim }{\to }{H}^{i+1}(M(\mathbb{Z}/n,i);\mathbb{Z}/n)$)

Attach an $(i+1)$-cell to $S^i$ by a degree-$n$ map, with $i\ge 1$ and $n\ge 2$. The integral cellular differential is multiplication by $n$, while its modulo-$n$ reduction is zero. Lifting the degree-$i$ generator modulo $n^2$ and applying the differential gives $n$, so the Bockstein homomorphism sends that generator to the degree-$(i+1)$ generator. This supplies a nonzero example in every positive degree.