TITLE: Euclidean domain in which the quotient and remainder are always unique
QUESTION [5 upvotes]: Let $R$ be a Euclidean domain in which the quotient and remainder are always unique. Does it follow that the ring $R$ is either a field or a polynomial ring $F[X]$ for some field $F$?

REPLY [5 votes]: Yes, that characterization is correct, see M. A. Jodeit, Jr., Uniqueness in the division algorithm, Amer. Math. Monthly 74 (1967), 835–836.