The element with this property is uniquely identified and is called the inverse element of and is often noted as.
Items that are not units are often referred to as non- units .
The set of all units of a monoid, so
forms a group , the unit group of . Another common name for the unit group is .
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.
- is always a unit because .
- 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 .
- 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.
- 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 .
- For a unitary ring , the unit group in the matrix ring is the general linear group consisting of the regular matrices.
- In the ring of whole numbers there are only the units and .
- In the ring of whole Gaussian numbers there are the four units .
- There are an infinite number of units in the ring . It is and therefore everyone is for units.
- 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.
- Units in unitary rings are never zero divisors.
- Are units, then also and units. It follows that the unit group is actually a group.
- 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
- An element that meets the condition for an element is called a left unit .
- An element that meets the condition for an element is called legal entity .
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.
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 .
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).
let be the matrix in , which has only ones in the first upper secondary diagonal and only zeros otherwise:
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.
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.
- 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 .
- Karpfinger, Meyberg: Algebra 2013, p. 9
- Karpfinger, Meyberg: Algebra 2013, Lemma 2.4
- Karpfinger, Meyberg: Algebra 2013, March 13
- Karpfinger, Meyberg: Algebra 2013, September 14