> The difference between a euclidean domain and a principal ideal domain A Euclidean domain is an integral domain where the Euclidean algorithm works. For the

> The difference between a euclidean domain and a principal ideal domain A Euclidean domain is an integral domain where the Euclidean algorithm works. For the Euclidean algorithm to work, you need to be able to divide two elements and produce a remainder that's smaller than the divisor. So an ED requires a notion of "smallness" (the Euclidean norm) which interacts with division in a way that makes the Euclidean algorithm work (remainder is always smaller than divisor). A principal ideal domain is an integral domain where every ideal is principal (can be generated by one element). It can be useful for you to know that certain situations cannot happen, e.g. in a PID you can say, "Let I be an ideal of D, then I = <g>..." and do something with the generating element g. It lets you pass from an ideal to a single generating element in a proof, which may be a useful capability. The PID concept is also part of a taxonomy, since Euclidean domains ⊆ principal ideal domains.