You have at least three different objects that may be called "p-adic rings".

The first one is the ring of p-adic integers, usually denoted by $\mathbb Z_p$ or by $\mathbb J_p$. There is plenty of different (but of course all equivalent) definitions of this ring, I list here some of these:

1- $\mathbb Z_p$ is the inverse limit of the finite rings $\mathbb Z(p^n)$ with $n\in \mathbb N_+$ (the transition map from $\mathbb Z(p^{n+1})$ to $\mathbb Z(p^n)$ is the obvious projection). Another way to say this is to say that $\mathbb Z_p$ is the completion of the ring $\mathbb Z$ with respect to the maximal ideal $p\mathbb Z$, that is, the completion of $\mathbb Z$ in its $p$-adic topology;

2- $\mathbb Z_p$ is the endomorphism ring of the Abelian $p$-Prüfer group $\mathbb Z(p^{\infty})$. (the equivalence with the above definition can be seen using the fact that $\mathbb Z(p^{\infty})$ is the direct limit of the Abelian groups $\mathbb Z(p^n)$)

3- $\mathbb Z_p$ is the ring of formal sums $\sum_{i=0}^{\infty}n_i p^i$, with $n_i\in \{0,\dots,p-1\}$.

The second object I want to describe is the field $\mathbb Q_p$ of the $p$-adic numbers, which is just the field of fractions of $\mathbb Z_p$. Really, all what you need to obtain $\mathbb Q_p$ from $\mathbb Z_p$ is to make invertible the operation "multiplication by $p$". So $\Q_p$ can be defined to be the direct limit of following direct system

$$\mathbb Z_p\overset{p}{\longrightarrow}\mathbb Z_p\overset{p}{\longrightarrow}\dots\overset{p}{\longrightarrow}\mathbb Z_p\overset{p}{\longrightarrow}\dots\, ,$$

that is, $\mathbb Q_p=\bigcup_{n\to\infty}1/p^n \mathbb Z_p$. Using the description 3 of $\mathbb Z_p$ I gave above you can see $\mathbb Q_p$ as the ring of all formal sums $\sum_{i=-k}^{\infty}n_i p^i$, with $n_i\in \{0,\dots,p-1\}$ and $k\in \N_+$.

One can also introduce the $p$-adic norm in $\mathbb Q$ and define $\mathbb Q_p$ as the completion with respect to $\mathbb Q_p$.

FInally, the last object that may be called a $p$-adic ring is a (topological) ring which has properties that are similar to the properties of $\mathbb Z_p$. More precisely, a commutative ring $R$, equipped with the $p$-adic topology, that is, the linear ring topology which has a base of neighborhoods of $0$ of the form

$$\ldots \subset p^3 R \subset p^2 R \subset p R\, ,$$

is said to be a $p$-adic ring if the following conditions are satisfied:

(a) the residue ring $\overline{k}=R/pR$ is a perfect ring of characteristic $p$;

(b) the ring $R$ is Hausdorff and complete (for the $p$-adic topology).

(c) $p$ is not a zero-divisor of $R$.

(In general, a commutative ring with a linear ring topology induced by a filtration by ideals $...\subset I_2\subset I_1$, is said to be a $p$-ring if it satisfies the above conditions (a) with $\overline{k}=R/I_1$ and (b) for this topology. In particular any $p$-adic ring is a $p$-ring).

Dear Ms Virili, thank you for a very complete introduction. Could you explain better the difference between Z_Pinfinity and Z_p? Thanks in advance!

EDIT: now I see what could eventually create the confusion: let me remark that the Prüfer group is the DIRECT limit of the finite cyclic p-groups, while the p-adic integers are the INVERSE limit of such groups (at least its underlying group coincides with such limit, otherwise you need to consider also the ring structure on the Z(p^n) and take the limit in the category of rings).

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Well, by $\mathbb Z(p^\infty)$ I mean the p-Prüfer group while by $\mathbb Z_p$ I mean the ring of $p$-adic integers.

The p-Prüfer group is an Abelian group which can be characterized in various ways. One of these ways is the one I mentioned in my previous answer, that is, as a direct union of the cyclic $p$-groups $\mathbb Z(p^n)$.

Furthermore, $\mathbb Z(p^\infty)$ is a divisible torsion indecomposable Abelian group; in this direction, one can define the $p$-Prúfer group as the divisible hull of the simple $\mathbb Z(p)$. (NB, $\mathbb Z$ is a P.I.D. so divisible hulls coincide with injective envelopes).

Another characterization of $\mathbb Z(p^\infty)$ is as the $p$-torsion part of the torus $T=\mathbb R/\mathbb Z$. In particular, it is the subgroup of $p^n$-th roots of unity with $n\in\mathbb N$. A more algebraic description of this approach is as follows. Let $\mathbb Z[1/p]$ be the subring of $\mathbb Q$ (or if you prefer of $\mathbb R$) generated by $1/p$ (so as a generic element of $\mathbb Z[1/p]$ is of the form $a/p^n$ with $a\in\mathbb Z$ and $n\in\mathbb N$). Then $\mathbb Z(p^\infty)$ is isomorphic to the quotient group $\mathbb Z[1/p]/\mathbb Z$.

I hope this clarifies the difference between $\mathbb Z(p^\infty)$ and $\mathbb Z_p$. The relation between the two things is the isomorphism

$$End(\mathbb Z(p^\infty)) \cong \mathbb Z_p .$$