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 .
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
Every finite multiplicative subgroup of a commutative field is cyclic (see root of unity ).
Examples
- The unit group of the ring of integers consists of the two elements 1 and −1.
- The unit group of the ring of rational numbers consists of all rational numbers not equal to zero, so it is a body.
- 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.
- 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.
- 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 .
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
- ↑ Andreas Bartholomé, Josef Rung, Hans Kern: Number theory for beginners. Vieweg + Teubner, 7th edition, 2010, page 113.