I've been studying about the division algorithm and found out that an euclidean domain may or may not have a division algorithm, but I've not found any examples of such domains.
Are they rare?
Can we put some condition on the ring $R$ in a way such that we have a division algorithm, for example if the quotient and remainder $(q,r)$ of the euclidean property are unique then $R$ is either isomorphic to a field or to the univariate polynomials over a field, and for those two we have division algorithms.
But in the case of $\mathbb{Z}$ we have that using the euclidean function $(q,r)$ are not unique (if we ask $0\leq r<|b|$ then they are but it is not the same condition), but $\mathbb{Z}$ does have a division algorithm!
Im not too sure if using the phrase "divison algorithm" is specific enough because I dont have any definition in mind for it but I guess it would be a deterministic algorithm that gives $(q,r)$ (which may not be an unique pair) in finitely many steps.
Thanks in advance.