Ring theory MOC

Euclidean domain

A Euclidean domain is an integral domain with a generalized version of the Euclidean division algorithm. More precisely, an integral domain 𝑅 is called a Euclidean domain iff there exists a Euclidean function 𝑑 :𝑅 →ℤ such that¹ ring

1. 0 ≤𝑑(𝑎) ≤𝑑(𝑎𝑏) for all nonzero 𝑎,𝑏 ∈𝐷; and
2. if 𝑎,𝑏 ∈𝐷 and 𝑏 ≠0, then there exist elements 𝑞,𝑟 ∈𝐷 such that 𝑎 =𝑞𝑏 +𝑟 and 𝑑(𝑟) <𝑑(𝑏).

Every Euclidean domain is a Principal ideal domain.

Proof

proof

Properties

• The polynomial ring over a field is a Euclidean domain

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Footnotes

1. 2017. Contemporary abstract algebra, §18, p. 315. ↩