Ring theory MOC

Unit

Let be a ring, and . Then

• is a left unit iff for some ;
• is a right unit iff for some ;
• 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 denote the multiplicative monoid of a ring viewed as a category. Then 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 and as functions on , then is¹

• a left unit iff is surjective iff is injective iff is not a right zero-divisor;

• a right unit iff is surjective iff is surjective iff 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 ↩