Back to the index. Or to the chambers

This article has 14 links. View as Cloud or List.

Proof Group Operation Multiplication Commutative Ring Ring Biconditional Field Inverse Number Congruences Units Of Quadratic Fields Prime Residue Class Congruence In Algebraic Number Field Gaussian Integers

Loading ...

Planetmath Browser (2008—2009)

BSD licence | A django site

All articles taken from PlanetMath.org snapshot under CC-BY-SA licence.

→ The original article on PlanetMath.org

Other Formats: LaTeX

New! You can click on formulas to copy the LaTeX source to your clipboard. | Math Videos

Group Of Units

Theorem 1 The set

of units of a ring

forms a group with respect to ring multiplication.

Proof. If and are two units, then there are the elements and of such that and . Then we get that , similarly . Thus also is a unit, which means that is closed under multiplication. Because $1 \in E$ and along with also its inverse belongs to , the set is a group.

Corollary. In a commutative ring, a ring product is a unit iff all factors are units.

Examples

When $R = \mathbb{Z}$ , then $E = \{1,\,-1\}$ .
When $R = \mathbb{Z}[i]$ , the ring of Gaussian integers, then $E = \{1,\,i,\,-1,\,-i\}$ .
When $R = \mathbb{Z}[\sqrt{3}]$ , then $E = \{\pm(2\!+\!\sqrt{3})^n\,\vdots\,\,\, n\in\mathbb{Z}\}$ .
When where is a field, then $E = K\!\smallsetminus\!\{0\}$ .
When $R = \{0\!+\!\mathbb{Z},\,1\!+\!\mathbb{Z},\,\ldots,\, m\!-\!1\!+\!\mathbb{Z}\}$ is the residue class ring modulo , then consists of the prime classes modulo , i.e. the residue classes $l\!+\!\mathbb{Z}$ satisfying $\gcd(l, m) = 1$ .

1	Field
1	Proof
1	Operation
1	Biconditional
1	Commutative Ring
1	Ring
1	Multiplication
1	Group
2	Inverse Number
3	Congruences
11	Units Of Quadratic Fields
11	Gaussian Integers
14	Prime Residue Class
15	Congruence In Algebraic Number Field