# 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 ${\ displaystyle M ^ {\ ast}}$

${\ displaystyle M ^ {\ ast}: = \ {x \ in M ​​\ mid x {\ text {is unit}} \},}$

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}$${\ displaystyle from}$${\ 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}$${\ displaystyle from = 1}$${\ 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 .

## 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