Local ring

A ring $𝑅$ is local¹ iff it is nontrivial and it has the property that for all $𝑎, 𝑏 \in 𝑅$ such that $𝑎 + 𝑏 = 1$ , either $𝑎$ or $𝑏$ is a unit. #m/def/ring/cons It follows that we have an apartness relation defined by

𝑎 # 𝑏 = 𝑎 - 𝑏 i s a u n i t .

Proof

Since $𝑅$ is nontrivial, $0$ is not a unit, giving irreflexivity.

Let $𝑥, 𝑦, 𝑧 : 𝑅$ , and suppose $𝑥 - 𝑧$ is a unit. Then $(𝑥 - 𝑦) + (𝑦 - 𝑧)$ is a unit, let $𝑎$ be its unique inverse. Then $(𝑥 - 𝑦) 𝑎 + (𝑦 - 𝑧) 𝑎 = 1$ so either $(𝑥 - 𝑦) 𝑎$ or $(𝑥 - 𝑦) 𝑎$ is a unit. If $(𝑥 - 𝑦) 𝑎$ is a unit, with inverse say $𝑏$ , then

(𝑥 - 𝑦) 𝑎 𝑏 = 1 = 𝑎 𝑏 (𝑥 - 𝑦)

so $𝑥 - 𝑦$ is a unit. The other case gives that $𝑦 - 𝑧$ is a unit. This proves cotransitivity.

Finally, note that if $𝑎 - 𝑏$ is a unit, then so too is $𝑏 - 𝑎 = - (𝑎 - 𝑏)$ , as the product of units. Thus symmetry also holds.

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

One would be justified to use the term “approximate division ring.” ↩

Local ring

Footnotes