Ring theory MOC

Unit

Let 𝑅 be a ring, and 𝑎 ∈𝑅. Then

• 𝑎 is a left unit iff 𝑎𝑏 =1 for some 𝑏;
• 𝑎 is a right unit iff 𝑏𝑎 =1 for some 𝑏;
• 𝑎 is a unit iff it is both a left unit and right unit.

By the usual argument, the inverse of an ambidextrous unit is unique, and these form the group of units. A ring in which every nonzero element is a unit is called a Division ring.

As morphisms

Let 𝑅―― denote the multiplicative monoid of a ring 𝑅 viewed as a category. Then 𝑎 ∈𝑅 is

• a left unit iff it is split epic;
• a right unit iff it is split monic;
• a unit iff it is an isomorphism.

If we view Λ(𝑎) and P(𝑎) as functions on 𝑅, then 𝑎 ∈𝑅 is¹

• a left unit iff Λ(𝑎) is surjective iff P(𝑎) is injective iff 𝑎 is not a right zero-divisor;

• a right unit iff P(𝑎) is surjective iff Λ(𝑎) is surjective iff 𝑎 is not a left zero-divisor;

See also

• Zero-divisor

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Footnotes

1. 2009. Algebra: Chapter 0,§III.1.2, ¶1.12, p. 123 ↩