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Calculate (find the type of isomorphism) of $\mathbb Z_p/p\mathbb Z_p$ where p is prime.

I know that $\mathbb Z/p\mathbb Z$ isomorphic to $\mathbb Z_p$ by showing

$\phi : \mathbb Z \rightarrow \mathbb Z_p$

$\phi(x) = x \bmod p$

$\ker \phi = p\mathbb Z$

and by isomorphism $1$ theorem.

So I get

$\mathbb Z_p/p\mathbb Z_p$ isomorphic to $(\mathbb Z/p\mathbb Z)/p(\mathbb Z/p\mathbb Z)$.

Not sure if I'm writing this in a correct way and if so then how to continue from here.

Any help and explanation will be appreciated.

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Assuming by $\mathbb Z_p$ you mean the mod $p$ number system rather than $p$-adics.

Another way to see this would be to realize that $p\mathbb Z_p$ is the $0$ subgroup (since $p = 0 \pmod p$).

Then $R/(0) = R$ so $\mathbb Z_p/p\mathbb Z_p = \mathbb Z_p$.

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