Ring theory MOC

GCD

Let 𝑅 be an integral domain and 𝑎,𝑏 ∈𝑅. An element 𝑑 ∈𝑅 is a greatest common divisor or GCD of 𝑎 and 𝑏 iff ring

⟨𝑎,𝑏⟩⊴⟨𝑑⟩

and ⟨𝑎,𝑏⟩ ⊴⟨𝑑′⟩ ⊴⟨𝑑⟩ implies ⟨𝑑′⟩ =⟨𝑑⟩.¹ The GCD is unique up to associate elements, leading to the abuse of notation

gcd{𝑎,𝑏}=𝑑.

Properties

1. GCDs exist for nonzero elements in a UFD

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Footnotes

1. 2009. Algebra: Chapter 0, §V.2.1, p. 252 ↩