Euclidean Domains

Definition 1   An integral domain $ R$ endowed with a function $ d: R \longmapsto {\mbox{${\mathbb{N}}$}} \ {\cup} \ \{-{\infty}\}$ is an Euclidean domain if the following two conditions hold The elements $ q$ and $ r$ are called the quotient and the remainder of $ a$ w.r.t. $ b$ (although $ q$ and $ r$ may not be unique). The function $ d$ is called the Euclidean size.

Example 1   Here are some classical examples.

Marc Moreno Maza