enhetsgruppe

Enhetsgruppen (unit group) of a ring R with unity is the set R^× consisting of all elements that have a multiplicative inverse in R. Under multiplication, R^× forms a group with identity element 1_R and inverse a^{-1} for every a in R^×. If R is commutative, the enhetsgruppe is an abelian group. Notation commonly used includes U(R) or R^×.

Typical examples illustrate the concept. In the ring of integers Z, the only units are 1 and -1. In the quotient ring Z/nZ, the units are the residue classes [a] with gcd(a, n) = 1; this unit group is denoted (Z/nZ)^× and has φ(n) elements, where φ is Euler’s totient function. For a field F, every nonzero element is a unit, so F^× equals F \ {0}. In the ring of n-by-n matrices M_n(R), the enhetsgruppe is the general linear group GL_n(R), consisting of all invertible matrices.

Key properties and notes. The enhetsgruppe is always a group under multiplication. In finite rings, R^× is finite. For finite commutative rings, the enhetsgruppe is abelian. In number theory, unit groups carry important arithmetic information: Dirichlet’s unit theorem describes the structure of the unit group of the ring of integers in a number field as a finitely generated abelian group, isomorphic to Z^r × T for some r ≥ 0 and finite T. Unit groups also arise in broader contexts such as algebraic geometry and linear algebra, where they correspond to invertible elements and maps.

Overall, the enhetsgruppe captures the invertible elements of a ring and forms a foundational object in algebra and its applications.