In the context of Gaussian integers, there can be non-unique quotients and their corresponding remainder, as for the example at page 6 of this, that states example of dividend($a$) of $3+2i$, divisor($b$) of $-1+3i$, and states that as long as the norm of the remainder is less than that of the norm of divisor. The example has dividend ($a$) as : $3 + 2i$ with norm = $13$, divisor($b$) as : $-1 + 3i$ with norm = $10$, and multiplying the fraction with conjugate of the denominator (divisor) yielding : $\frac{3}{10} + i\frac{-11}{10}$.
The first set of values derived is : $q = -i, r = i$.
The alternate value set suggested is: $q = 1-i$, $r= 1-2i$, as the norm of remainder $\lt$ than that of divisor.
$(-1 + 3i)(1-i) + (1-2i) \implies (-1 +4i +4) + (1 -2i) = 3 +2i$.
I have two doubts:
I want to know the significance of the different solution sets of (quotient, remainder), say how they are located on the lattice w.r.t. each other.
Can there be an infinity of quotients and remainders for any possible given set of dividend ($a$) and divisor ($b$) in algebraic integers, if $N_a \gt N_b$. If yes, then is it also true that the condition for such solution sets' finding is nothing except $N_a \gt N_b$.
Suppose $\alpha$ and $\beta\ne0$ are algebraic integers in some ring. The result of dividing $\frac{\alpha}{\beta}=\gamma$ is an algebraic number, which in general, may not be an integer. If we locate this algebraic number in the lattice, and then look at the set of algebraic integers $q$ satisfying $N(q\alpha-\gamma)<N(\beta)$, we find possible quotients that will "work" in the division algorithm.
If we are in a ring where we can show that this set of possible quotients is always non-empty, then we are in a ring where the division algorithm "works", i.e., a Euclidean domain. Not every ring of algebraic integers is a Euclidean domain, but in the ones that are, we have a lot of nice results. In a ring without a working division algorithm, other techniques are necessary.