Unit (math)

In algebra , a branch of mathematics , an invertible element of a monoid is called a unit . Units are mainly considered in unitary rings .

definition

Let be a monoid , where with denotes the neutral element . Then an element is called a unit if it is invertible, i.e. if there is one with ${\displaystyle (M,\cdot ,1)}$ ${\displaystyle 1}$ ${\displaystyle a\in M}$ ${\displaystyle b\in M}$

${\displaystyle a\cdot b=b\cdot a=1}$.

The element with this property is uniquely identified and is called the inverse element of and is often noted as. ${\displaystyle b}$ ${\displaystyle a}$ ${\displaystyle a^{-1}}$

Items that are not units are often referred to as non- units .

The set of all units of a monoid, so ${\displaystyle M^{\ast }}$

${\displaystyle M^{\ast }:=\{x\in M\mid x{\text{ ist Einheit}}\},}$

forms a group , the unit group of . Another common name for the unit group is . ${\displaystyle M}$ ${\displaystyle M^{\times }}$

Special case: units in unitary rings

Let be a unitary ring , i.e. a ring with a neutral element with respect to the multiplication, which is denoted by. Then is a monoid and thus the concept of unit is defined for a unitary ring and is precisely the set of invertible elements. ${\displaystyle (R,+,\cdot ,0,1)}$ ${\displaystyle 1}$ ${\displaystyle (R,\cdot ,1)}$

Examples

• ${\displaystyle 1}$is always a unit because . ${\displaystyle 1\cdot 1=1}$
• ${\displaystyle 0}$is a unit in a ring if and only if the ring is the zero ring .
• Is in a body . This means that apart from the 0, every element in a body is a unit. In general, rings other than the zero ring, in which all elements are units, are called inclined bodies . ${\displaystyle K}$ ${\displaystyle K^{*}=K\setminus \{0\}}$ ${\displaystyle 0}$
• In the polynomial ring above an integrity ring, the following applies . In particular, for a body one obtains that . The units here correspond exactly to the polynomials with degree zero. ${\displaystyle R}$ ${\displaystyle R[X]^{*}\cong R^{*}}$ ${\displaystyle \mathbb {K} }$ ${\displaystyle \mathbb {K} [X]^{*}\cong \mathbb {\mathbb {K} } \setminus \{0\}}$
• The units in the ring of formal power series above a commutative ring are exactly those power series whose absolute term is a unit in . ${\displaystyle R[[X]]}$ ${\displaystyle R}$ ${\displaystyle a_{0}}$ ${\displaystyle R}$
• For a unitary ring , the unit group in the matrix ring is the general linear group consisting of the regular matrices. ${\displaystyle R}$ ${\displaystyle R^{n\times n}}$ ${\displaystyle GL(n,R)}$
• In the ring of whole numbers there are only the units and . ${\displaystyle \mathbb {Z} }$ ${\displaystyle 1}$ ${\displaystyle -1}$
• In the ring of whole Gaussian numbers there are the four units . ${\displaystyle \mathbb {Z} [\mathrm {i} ]}$ ${\displaystyle 1,-1,i,-i}$
• There are an infinite number of units in the ring . It is and therefore everyone is for units. ${\displaystyle \mathbb {Z} [\mathrm {\sqrt {3}} ]}$ ${\displaystyle \left(2+{\sqrt {3}}\right)\left(2-{\sqrt {3}}\right)=1}$ ${\displaystyle \left(2+{\sqrt {3}}\right)^{k}}$ ${\displaystyle k\in \mathbb {N} }$
• The last two rings are examples of integral rings of square number fields . The producers of the unit group are known for these. Using more general number fields , Dirichlet's unit theorem makes a weaker statement about the structure of the units.

properties

• Units in unitary rings are never zero divisors.
• Are units, then also and units. It follows that the unit group is actually a group. ${\displaystyle a,b\in M}$ ${\displaystyle ab}$ ${\displaystyle a^{-1}}$
• Finite subsets of the unit group of an integrity ring are always cyclic.
• Every non-unit of a commutative unitary ring lies in a maximal ideal . In particular, the unit group is the complement of the union of all maximum ideals and a ring only has one maximum ideal, i.e. it is a local ring , if the non-units form an ideal.

Generalization: left and right units

If the monoid is not commutative, one-sided units can also be considered ${\displaystyle M}$

• An element that meets the condition for an element is called a left unit . ${\displaystyle a\in M}$ ${\displaystyle ab=1}$ ${\displaystyle b\in M}$
• An element that meets the condition for an element is called legal entity . ${\displaystyle a\in M}$ ${\displaystyle ba=1}$ ${\displaystyle b\in M}$

An element is a unit if and only if it is a left unit and a right unit at the same time. In a commutative monoid, the three terms match. remains a bilateral unity even in the non-commutative case. ${\displaystyle a\in M}$ ${\displaystyle 1}$

example

There is the following ring in which there is a left unit that is not a right unit and a right unit that is not a left unit . In addition, and are still one-sided zero divisors . ${\displaystyle R}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle A}$ ${\displaystyle B}$

${\displaystyle R}$consists of all matrices of the size “countable-times-countable” with components in the real numbers , in which there are only finitely many non-zeros in each row and in each column (a total of infinitely many non-zeros may be included). is a ring with the usual matrix addition and matrix multiplication . The identity matrix has only ones on the main diagonal and zeros otherwise, it is the one element of (the neutral element of multiplication). ${\displaystyle R}$ ${\displaystyle E}$ ${\displaystyle R}$

${\displaystyle A}$let be the matrix in , which has only ones in the first upper secondary diagonal and only zeros otherwise: ${\displaystyle R}$

${\displaystyle A={\begin{pmatrix}0&1&0&0&0&\\0&0&1&0&0&\cdots \\0&0&0&1&0&\\0&0&0&0&1&\ddots \\&\vdots &&&\ddots &\ddots \end{pmatrix}}}$

Let be the transpose of , or in words the matrix that has only ones in the first diagonal below the main diagonal, and only zeros otherwise. ${\displaystyle B=A^{\mathrm {T} }}$ ${\displaystyle A}$

It is , so is a left unity and a right unity. For each element of , however, the product in the first column has only zeros and the product in the first row has only zeros. There can therefore be no legal entity and no left entity . With the matrix , which only contains a one in the component and otherwise only zeros, is and , so is a left zero divisor and a right zero divisor. ${\displaystyle AB=E}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle C}$ ${\displaystyle R}$ ${\displaystyle CA}$ ${\displaystyle BC}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle D}$ ${\displaystyle D_{1,1}}$ ${\displaystyle AD=0}$ ${\displaystyle DB=0}$ ${\displaystyle A}$ ${\displaystyle B}$

A functional analysis variant of this example is the unilateral shift operator .

literature

• Siegfried Bosch : Algebra. 8th edition. Springer, Berlin / Heidelberg 2013, ISBN 978-3-642-39566-6 .
• Gerd Fischer : Textbook of Algebra. 3. Edition. Springer, Wiesbaden 2013, ISBN 978-3-658-02220-4 , p. 147.
• Jens Carsten Jantzen , Joachim Schwermer : Algebra. 2nd Edition. Springer, Berlin / Heidelberg 2014, ISBN 978-3-642-40532-7 , III, §2.
• Christian Karpfinger , Kurt Meyberg: Algebra. 3. Edition. Springer, Berlin / Heidelberg 2013, ISBN 978-3-8274-3011-3 .

Individual evidence

1. ↑ Karpfinger, Meyberg: Algebra 2013, p. 9
2. ↑ Karpfinger, Meyberg: Algebra 2013, Lemma 2.4
3. ↑ Karpfinger, Meyberg: Algebra 2013, March 13
4. ↑ Karpfinger, Meyberg: Algebra 2013, September 14