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In mathematics, a 'unit' in a (unital) ring ''R'' is an invertible element of ''R'', i.e. an element ''u'' such that there is a ''v'' in ''R'' with
:''uv'' = ''vu'' = 1''R'', where 1''R'' is the multiplicative identity element.
That is, ''u'' is an ''invertible'' element of the multiplicative monoid of ''R''.
Unfortunately, the term ''unit'' is also used to refer to the identity element 1''R'' of the ring, in expressions like ''ring with a unit'' or ''unit ring'', and also e.g. '''unit' matrix''. (For this reason, some authors call 1'R' "unity", and say that ''R'' is a "ring with unity" rather than "ring with a unit". Note also that the term ''unit matrix'' more usually denotes a matrix with all elements equal to one.)
The units of ''R'' form a group ''U''(''R'') under multiplication, the 'group of units' of ''R''. The group of units ''U''(''R'') is sometimes also denoted ''R''
★ or ''R''×.
In a commutative unital ring ''R'', the group of units ''U''(''R'') acts on ''R'' via multiplication. The orbits of this action are called sets of ''associates''; in other words, there is an equivalence relation ~ on ''R'' called ''associatedness'' such that
:''r'' ~ ''s''
means that there is a unit ''u'' with ''r'' = ''us''.
One can check that ''U'' is a functor from the category of rings to the category of groups: every ring homomorphism ''f'' : ''R'' → ''S'' induces a group homomorphism ''U''(''f'') : ''U''(''R'') → ''U''(''S''), since ''f'' maps units to units. This functor has a left adjoint which is the integral group ring construction.
In an integral domain the cardinality of an equivalence class of associates is the same as that of ''U''(''R'').
A ring ''R'' is a field if and only if ''R''
★ = ''R'' {0}.
★ In the ring of integers, 'Z', the units are ±1. The associates are pairs ''n'' and −''n''.
★ Any root of unity is a unit in any unital ring ''R''. (If ''r'' is a root of unity, and ''r''''n'' = 1, then ''r''−1 = ''r''''n'' − 1 is also an element of ''R'' by closure under multiplication.) In algebraic number theory, Dirichlet's unit theorem shows the existence of many units in most rings of algebraic integers. For example, we have (√5 + 2)(√5 − 2) = 1.
★ In the ring ''M''(''n'','F') of ''n''×''n'' matrices over some field 'F' the units are exactly the invertible matrices.
:''uv'' = ''vu'' = 1''R'', where 1''R'' is the multiplicative identity element.
That is, ''u'' is an ''invertible'' element of the multiplicative monoid of ''R''.
Unfortunately, the term ''unit'' is also used to refer to the identity element 1''R'' of the ring, in expressions like ''ring with a unit'' or ''unit ring'', and also e.g. '''unit' matrix''. (For this reason, some authors call 1'R' "unity", and say that ''R'' is a "ring with unity" rather than "ring with a unit". Note also that the term ''unit matrix'' more usually denotes a matrix with all elements equal to one.)
| Contents |
| Group of units |
| Examples |
Group of units
The units of ''R'' form a group ''U''(''R'') under multiplication, the 'group of units' of ''R''. The group of units ''U''(''R'') is sometimes also denoted ''R''
★ or ''R''×.
In a commutative unital ring ''R'', the group of units ''U''(''R'') acts on ''R'' via multiplication. The orbits of this action are called sets of ''associates''; in other words, there is an equivalence relation ~ on ''R'' called ''associatedness'' such that
:''r'' ~ ''s''
means that there is a unit ''u'' with ''r'' = ''us''.
One can check that ''U'' is a functor from the category of rings to the category of groups: every ring homomorphism ''f'' : ''R'' → ''S'' induces a group homomorphism ''U''(''f'') : ''U''(''R'') → ''U''(''S''), since ''f'' maps units to units. This functor has a left adjoint which is the integral group ring construction.
In an integral domain the cardinality of an equivalence class of associates is the same as that of ''U''(''R'').
A ring ''R'' is a field if and only if ''R''
★ = ''R'' {0}.
Examples
★ In the ring of integers, 'Z', the units are ±1. The associates are pairs ''n'' and −''n''.
★ Any root of unity is a unit in any unital ring ''R''. (If ''r'' is a root of unity, and ''r''''n'' = 1, then ''r''−1 = ''r''''n'' − 1 is also an element of ''R'' by closure under multiplication.) In algebraic number theory, Dirichlet's unit theorem shows the existence of many units in most rings of algebraic integers. For example, we have (√5 + 2)(√5 − 2) = 1.
★ In the ring ''M''(''n'','F') of ''n''×''n'' matrices over some field 'F' the units are exactly the invertible matrices.
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