I was reading Proposition 3.9 on p. 121 of the book Algebraic Number Theory by Jürgen Neukirch. It said:
If $v$ is a discrete exponential valuation of $K$, then $$ \mathcal{o}=\{x\in K|v(x)\ge0\} $$ is a PID, hence a DVR. Suppose $v$ is normalized. Then the nonzero ideals of $\mathcal{o}$ are $$ \mathfrak{p}^n=\pi^n\mathcal{o}=\{x\in K|v(x)\ge n\},\quad n\ge0, $$ where $\pi$ is a prime element, i.e., $v(\pi)=1$. One has $$ \mathfrak{p}^n/\mathfrak{p}^{n+1}\cong\mathcal{o}/\mathfrak{p}. $$
Please let me ask a few questions:
Is "prime element" here synonymous with prime elements of (the commutative ring) $\mathcal{o}$?
The isomorphism $\mathfrak{p}^n/\mathfrak{p}^{n+1}\cong\mathcal{o}/\mathfrak{p}$ being written, I guess this does not refer to isomorphism of rings, does it? Or just of the underlying additive groups?
Thank you very much for answering.