What can be said about the relation between $\mathbb{Z}_p$ and $\mathbb{Z}_{(p)}$?

Question

What can be said about the relation between $\mathbb{Z}_p$ and $\mathbb{Z}_{(p)}$? This is a question in Hungerford.

I understand what both are, $\mathbb{Z}_p = \mathbb{Z}/(p)$ is a finite field and $\mathbb{Z}_{(p)} = S^{-1}\mathbb{Z}$ where $S = \mathbb{Z}-(p)$. Also $\mathbb{Z}_{(p)}$ is a subring of $\mathbb{Q}$ consisting of quotients $\frac mn$ where $n$ does not divide $p$. But I'm not sure what the question is asking for. Is there some nice relations between the two?

Answer 1

Assuming $p$ is prime.

Well, the image of $S$ in $\mathbb{Z}/p\mathbb{Z}$ consists of units, so by the universal property of localization there is a homomorphism $\lambda: \mathbb{Z}_{(p)} \rightarrow \mathbb{Z}/p\mathbb{Z}$ such that $\lambda(x) = x + p\mathbb{Z}$ for all $x \in \mathbb{Z}$. This mapping is surjective, and it maps $m/n$ to $mn^{-1} + p\mathbb{Z}$. So the kernel is the image of $p\mathbb{Z}$ in the localization.

Thus $\mathbb{Z}_{(p)} / p\mathbb{Z}_{(p)} \cong \mathbb{Z}/p\mathbb{Z}$.

More generally when $P$ is a prime ideal of a commutative ring $R$, then $R_P / P_P$ is isomorphic to the field of fractions of $R/P$.

Answer 2

Usually in this context $\mathbb{Z}_p$ denotes the $p$-adic integers, and $\mathbb{Z}/(p)$ the quotient ring. So my answer refers to $p$-adic integers. One nice relationship then is, that the p-adic integers are the completion of the $p$-local numbers as valuation ring, i.e., we have $$ \widehat{\mathbb{Z}_{(p)}}=\mathbb{Z}_p. $$