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.