Let R be a ring with multiplicative identityPlanetmathPlanetmath 1. We say that uR is an unit (or unital) if u divides 1 (denoted u1). That is, there exists an rR such that 1=ur=ru.

Notice that r will be the multiplicative inverse (in the ring) of u, so we can characterize the units as those elements of the ring having multiplicative inverses.

In the special case that R is the ring of integersMathworldPlanetmath of an algebraic number fieldMathworldPlanetmath K, the units of R are sometimes called the algebraic units of K (and also the units of K).  They are determined by Dirichlet’s unit theorem.

Title unit
Canonical name Unit
Date of creation 2013-03-22 11:56:28
Last modified on 2013-03-22 11:56:28
Owner drini (3)
Last modified by drini (3)
Numerical id 15
Author drini (3)
Entry type Definition
Classification msc 16B99
Synonym unital
Related topic AssociatesMathworldPlanetmath
Related topic Prime
Related topic Ring
Related topic UnitsOfQuadraticFields
Defines algebraic unit