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Unit (ring theory)
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== Group of units == A [[commutative ring]] is a [[local ring]] if {{math|''R'' ∖ ''R''{{sup|Γ}}}} is a [[maximal ideal]]. As it turns out, if {{math|''R'' ∖ ''R''{{sup|Γ}}}} is an ideal, then it is necessarily a [[maximal ideal]] and {{math|''R''}} is [[local ring|local]] since a [[maximal ideal]] is disjoint from {{math|''R''{{sup|Γ}}}}. If {{mvar|R}} is a [[finite field]], then {{math|''R''{{sup|Γ}}}} is a [[cyclic group]] of order {{math|{{abs|''R''}} β 1}}. Every [[ring homomorphism]] {{math|''f'' : ''R'' β ''S''}} induces a [[group homomorphism]] {{math|''R''{{sup|Γ}} β ''S''{{sup|Γ}}}}, since {{mvar|f}} maps units to units. In fact, the formation of the unit group defines a [[functor]] from the [[category of rings]] to the [[category of groups]]. This functor has a [[left adjoint]] which is the integral [[group ring]] construction.{{sfn|Cohn|2003|loc=Β§2.2 Exercise 10|ps=}} The [[group scheme]] <!-- shouldn't we avoid scheme? --><math>\operatorname{GL}_1</math> is isomorphic to the [[multiplicative group scheme]] <math>\mathbb{G}_m</math> over any base, so for any commutative ring {{mvar|R}}, the groups <math>\operatorname{GL}_1(R)</math> and <math>\mathbb{G}_m(R)</math> are canonically isomorphic to {{math|''U''(''R'')}}. Note that the functor <math>\mathbb{G}_m</math> (that is, {{math|''R'' β¦ ''U''(''R'')}}) is [[Representable functor|representable]] in the sense: <math>\mathbb{G}_m(R) \simeq \operatorname{Hom}(\mathbb{Z}[t, t^{-1}], R)</math> for commutative rings {{mvar|R}} (this for instance follows from the aforementioned adjoint relation with the group ring construction). Explicitly this means that there is a natural bijection between the set of the ring homomorphisms <math>\mathbb{Z}[t, t^{-1}] \to R</math> and the set of unit elements of {{mvar|R}} (in contrast, <math>\mathbb{Z}[t]</math> represents the additive group <math>\mathbb{G}_a</math>, the [[forgetful functor]] from the category of commutative rings to the [[category of abelian groups]]).
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