Definition of Euclidean Domain and Principal Ideal Domain (PID)

A Euclidean domain is an integral domain R with a function d:R\setminus \{0\}\to \mathbb{N} satisfying the following:

(1) d(a)\leq d(ab) for all nonzero a,b in R.

(2) for all a,b \in R, b\neq 0, \exists q, r, \in R such that a=bq+r, with either r=0 or d(r)<d(b).

(d is known as the Euclidean function)

On the other hand, a Principal ideal domain (PID) is an integral domain in which every ideal is principal (can be generated by a single element).

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2 thoughts on “Definition of Euclidean Domain and Principal Ideal Domain (PID)”

  1. Pingback: Proof that a Euclidean Domain is a PID (Principal Ideal Domain) | Singapore Maths Tuition
  2. Pingback: Z[Sqrt(-2)] is a Principal Ideal Domain Proof | Singapore Maths Tuition

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