In what sense $f_n:\mathbb{Z}/p^{n+1} \to \mathbb{Z}/p^n \mathbb{Z}$ is a projection map?

Question

Why the map $\mathbb{Z}/p^{n+1} \mathbb{Z} \to \mathbb{Z}/p^n \mathbb{Z}$ in the definition of p-adic ring of integer $\mathbb{Z}_p$ is called projection map ?

The definition is taken by inverse limit as follows:

$\mathbb{Z}_p=\varprojlim \mathbb{Z}/p^n \mathbb{Z},$ where homomorphisms are the projection maps $f_n:\mathbb{Z}/p^{n+1} \to \mathbb{Z}/p^n \mathbb{Z}$.

But I could not understand the projection map $f_n:\mathbb{Z}/p^{n+1} \to \mathbb{Z}/p^n \mathbb{Z}$.

In what sense $f_n:\mathbb{Z}/p^{n+1} \to \mathbb{Z}/p^n \mathbb{Z}$ is a projection map? what is the projected component in image set?

Answer 1

It could be thought of as a projection in the following sense. An element of $\mathbb Z /p^n \mathbb Z$ can be thought of as a number written in base $p$ with $n$ digits (including leading zeroes). Then arithmetic -- at least $+$ and $\times$ -- on these numbers is like ordinary arithmetic, except that you throw away any digits in a position higher than $n$ that appear.

Then this mapping "projects" $n+1$-digit numbers onto the last $n$ digits. For example, when $p = 5$ and $n = 4$, the mapping will take $21410$ to $1410$.

Answer 2

Whereas Mees de Vries' answer is indeed a good way to think about this specific situation, and whereas Lord Shark the Unknown's comment is indeed a very general way to approach this, here is somewhat of a golden mean to see this:

One of the isomorphism theorems (in Wikipedia's numeration, the third) says that

If $T \subseteq S \subseteq M$ are $R$-modules, then the quotient module $(M/T)/(S/T)$ is isomorphic to $M/S$.

In our situation $R=\mathbb Z$, $M=\mathbb Z$, $S=p^n \mathbb Z$, $T=p^{n+1} \mathbb Z$, this isomorphism identifies the "actual" projection -- if by that, we mean a quotient map --

$$\mathbb Z/p^{n+1}\mathbb Z \twoheadrightarrow (\mathbb Z/p^{n+1}\mathbb Z)/(p^n \mathbb Z/p^{n+1}\mathbb Z)$$

with the composite map

$$\mathbb Z/p^{n+1}\mathbb Z \twoheadrightarrow (\mathbb Z/p^{n+1}\mathbb Z)/(p^n \mathbb Z/p^{n+1}\mathbb Z) \simeq \mathbb Z/p^{n}\mathbb Z.$$