Associate elements

Let $𝑅$ be a ring. Two elements $𝑎, 𝑏 \in 𝑅$ are associate iff their corresponding principal ideals are equal, i.e. $⟨ 𝑎 ⟩ = ⟨ 𝑏 ⟩$ . For $𝑅$ an integral domain, this is equivalent to the existence of a unit $𝑢 \in 𝑅 \times$ such that $𝑎 = 𝑢 𝑏$ .

Proof of equivalence in a domain

Assume $⟨ 𝑎 ⟩ = ⟨ 𝑏 ⟩$ , so there exist $𝑐, 𝑑 \in 𝑅$ such that $𝑏 = 𝑎 𝑐$ and $𝑎 = 𝑏 𝑑$ so $𝑎 = 𝑏 𝑑 = 𝑎 𝑐 𝑑$ and thus $𝑎 (1 - 𝑐 𝑑) = 0$ , so $𝑐 𝑑 = 1$ and thus $𝑐 \in 𝑅 \times$ .

Conversely, if $𝑎 = 𝑢 𝑏$ , then $𝑏 = 𝑢 - 1 𝑎$ , so $⟨ 𝑎 ⟩ = ⟨ 𝑏 ⟩$ .

#state/tidy | #lang/en | #SemBr