Endomorphism ring of $p$-adic integer $Z_{p}$

Question

I would like to figure out the endomorphism ring of p-adic integer $Z_{p}$.

I know $End（Z）$ is isomorphic to $Z$,

And I guess $End（Zp）$ is also $Zp$.

My image : End$（Z_{p}）$$＝End（limZ/p^nZ）＝lim （End（Z/p^nZ））$$＝limZ/p^nZ＝$$Z_{p}$

My image might be wrong, I want to formally prove that the endomorphsm ring of $Zp$ is $Zp$. Thank you in advance.

Answer 1

There is an inclusion $\mathbb Z_p \to \operatorname{End}_{\mathbb Z_p}(\mathbb Z_p)$, which sends $a$ to multiplication by $a$.

You are correct that this map is surjective: Let $f$ be an endomorphism and let $a = f(1)$. By $\mathbb Z_p$-linearity, we then have $f(b \cdot 1) = b \cdot f(1) = ab$ for all $b \in \mathbb Z_p$. Thus $f$ is multiplication by $a$.

The same proof works for any commutative ring (with 1): every $A$-module endomorphism of $A$ is multiplication by some $a \in A$.