calculate (find the type of isomorphism) of $ℤ_{p^n}/pℤ_{p^n}$ where p is prime.

Question

calculate (find the type of isomorphism) of $ℤ_{p^n}/pℤ_{p^n}$ where p is prime.

I know that $ℤ/pℤ$ isomorphic to $ℤ_p$

and that $ℤ_p/pℤ_p$ isomorphic to $Z_p$

any help and explanation on how to continue will be appreciated

Answer 1

The ideals in a quotient $A/\mathfrak{a}$ correspond bijectively with the ideals in $A$ which contain $\mathfrak{a}$. In this way $\mathbb{Z}_{p^n}/ p\mathbb{Z}_{p^n}$ is isomorphic to $\mathbb{Z}/((p)+(p^n)) = \mathbb{Z}/(p) = \mathbb{Z}_p$.