# Unit group

In mathematics , the unit group of a ring with one element is the set of all multiplicatively invertible elements. With the ring multiplication it is a group .

The unit groups of (unitary) associative algebras can be seen as generalizations of the general linear group .

## definition

Let be a ring with 1. The set of all multiplicatively invertible elements ( units ) of forms a group with the ring multiplication . It is called the unit group of . The unit group is usually written as or as . The definition can be transferred to monoids . ${\ displaystyle R}$${\ displaystyle R}$${\ displaystyle R}$${\ displaystyle R ^ {*}}$${\ displaystyle R ^ {\ times}}$

## Properties and related terms

• A commutative ring with 1, whose unit group consists of all elements except zero, is already a body .
• A commutative ring with 1 is local if and only if the complement of the unit group is an ideal .

## The unit group of a body

The unit group of a body is called a multiplicative group. It is isomorphic to the linear algebraic group${\ displaystyle K ^ {*} = \ {x \ in K \ mid x \ neq 0 \}}$${\ displaystyle K}$

${\ displaystyle \ mathbb {G} _ {m} (K): = \ left \ {{\ begin {pmatrix} x & 0 \\ 0 & ~ x ^ {- 1} \ end {pmatrix}} \; {\ Bigg | } \; x \ in K ^ {*} \ right \} \ subseteq \ mathrm {GL} _ {2} (K).}$

Every finite multiplicative subgroup of a commutative field is cyclic (see root of unity ). ${\ displaystyle K}$

## Examples

• The unit group of the ring of integers consists of the two elements 1 and −1.${\ displaystyle \ mathbb {Z}}$
• The unit group of the ring of rational numbers consists of all rational numbers not equal to zero, so it is a body.${\ displaystyle \ mathbb {Q}}$${\ displaystyle \ mathbb {Q}}$
• The unit group of the remainder class ring modulo 10 consists of elements 1, 3, 7 and 9.
• If a number is prime, then there are exactly units.${\ displaystyle p}$${\ displaystyle \ mathbb {Z} / p \ mathbb {Z}}$${\ displaystyle p-1}$
• General: Is , so there are in exactly units. Here is the Euler function . is the number of natural numbers that do not greater than and relatively prime to have.${\ displaystyle m \ in \ mathbb {N}}$${\ displaystyle \ mathbb {Z} / m \ mathbb {Z}}$${\ displaystyle \ varphi (m)}$${\ displaystyle \ varphi}$${\ displaystyle \ varphi (m)}$${\ displaystyle m}$${\ displaystyle m}$
• The unit group of the matrix ring of the matrices with coefficients in a body is called the general linear group . and are lie groups .${\ displaystyle n \ times n}$${\ displaystyle K}$ ${\ displaystyle \ mathrm {GL} (n, K)}$${\ displaystyle \ mathrm {GL} (n, \ mathbb {R})}$${\ displaystyle \ mathrm {GL} (n, \ mathbb {C})}$

## literature

• Andreas Bartholomé, Josef Rung, Hans Kern: Number theory for beginners. Vieweg + Teubner, 7th edition, 2010, ISBN 978-3-8348-1213-1 .
• Armin Leutbecher: Number Theory. An introduction to algebra. Springer, Berlin / Heidelberg / New York 1996, ISBN 3-540-58791-8 .

## Individual evidence

1. Andreas Bartholomé, Josef Rung, Hans Kern: Number theory for beginners. Vieweg + Teubner, 7th edition, 2010, page 113.