Finding the residue field for the local ring $\mathbb{Z}_{\langle 5\rangle}$ with maximal ideal $5\mathbb{Z}_{\langle 5\rangle}$

Question

I wish to find the residue field for the local ring $\mathbb{Z}_{\langle 5\rangle}$ with maximal ideal $5\mathbb{Z}_{\langle 5\rangle}$, i.e. compute the quotient $\mathbb{Z}_{\langle 5\rangle}/5\mathbb{Z}_{\langle 5\rangle}$. It seems intuitively that this should be isomorphic to $\mathbb{Z}/5\mathbb{Z}= \mathbb{F}_5$, but I have no idea how to construct the isomorphism.

I tried by considering a map $\mathbb{Z}_{\langle 5\rangle}\rightarrow \mathbb{Z}/5\mathbb{Z}$, with a view to applying the First Isomorphism Theorem, which takes elements $\frac{a}{b}$ to $a $ mod $5$. However, I was struggling to show that this map is well-defined. Any help would be greatly appreciated!

Answer 1

$5\Bbb Z_5$ is the zero ideal. Thus we do get back the original $\Bbb Z_5$.