Let $(R,f)$ be an Euclidean domain with Euclidean function $f: R\setminus \lbrace 0 \rbrace \to \mathbb{Z}_+$. Given a fixed non-unit $b$ and $x \neq 0$, we obtain by division with remainder $x_0,y_1$ s.t. $x=x_0 + y_1b$. Repeating this process we find $x_0,...,x_n,y_{n+1}$ s.t. $x=x_0 + x_1b + \cdots + x_nb^n + y_{n+1}b^{n+1}$ with $x_i=0$ or $f(x_i) < f(b)$ for $i=0,...,n$.
a) Does this process necessarily terminate, i.e. has each $x\neq 0$ a $b$-adic represenation $x=x_0 + x_1b + \cdots + x_nb^n$ with $x_i=0$ or $f(x_i) < f(b)$ for $i=0,...,n$ ?
b) If not, what are counter-examples ?
c) If not, what are additional conditions on $f$ such that a) holds ?