Editing Adjoint functors (section)

==== Algebra ====
* '''Adjoining an identity to a [[Rng (algebra)|rng]].'''  This example was discussed in the motivation section above.  Given a rng ''R'', a multiplicative identity element can be added by taking ''R''x'''Z''' and defining a '''Z'''-bilinear product with (r,0)(0,1) = (0,1)(r,0) = (r,0), (r,0)(s,0) = (rs,0), (0,1)(0,1) = (0,1). This constructs a left adjoint to the functor taking a ring to the underlying rng.
* '''Adjoining an identity to a [[semigroup]].''' Similarly, given a semigroup ''S'', we can add an identity element and obtain a [[monoid]] by taking the [[disjoint union]] ''S'' <math>\sqcup</math> {1} and defining a binary operation on it such that it extends the operation on ''S'' and 1 is an identity element. This construction gives a functor that is  a left adjoint to the functor taking a monoid to the underlying semigroup.
* '''Ring extensions.''' Suppose ''R'' and ''S'' are rings, and ρ : ''R'' → ''S'' is a [[ring homomorphism]]. Then ''S'' can be seen as a (left) ''R''-module, and the [[tensor product]] with ''S'' yields a functor ''F'' : ''R''-'''Mod''' → ''S''-'''Mod'''. Then ''F'' is left adjoint to the forgetful functor ''G'' : ''S''-'''Mod''' → ''R''-'''Mod'''.
* '''[[Tensor-hom adjunction|Tensor products]].''' If ''R'' is a ring and ''M'' is a right ''R''-module, then the tensor product with ''M'' yields a functor ''F'' : ''R''-'''Mod''' → '''Ab'''. The functor ''G'' : '''Ab''' → ''R''-'''Mod''', defined by ''G''(''A'') = hom<sub>'''Z'''</sub>(''M'',''A'') for every abelian group ''A'', is a right adjoint to ''F''.
* '''From monoids and groups to rings.''' The [[integral monoid ring]] construction gives a functor from [[monoid]]s to rings. This functor is left adjoint to the functor that associates to a given ring its underlying multiplicative monoid. Similarly, the [[integral group ring]] construction yields a functor from [[group (mathematics)|groups]] to rings, left adjoint to the functor that assigns to a given ring its [[group of units]]. One can also start with a [[field (mathematics)|field]] ''K'' and consider the category of ''K''-[[associative algebra|algebras]] instead of the category of rings, to get the monoid and group rings over ''K''.
* '''Field of fractions.''' Consider the category '''Dom'''<sub>m</sub> of integral domains with injective morphisms. The forgetful functor '''Field''' → '''Dom'''<sub>m</sub> from fields has a left adjoint—it assigns to every integral domain its [[field of fractions]].
* '''Polynomial rings'''. Let '''Ring'''<sub>*</sub> be the category of pointed commutative rings with unity (pairs (A,a) where A is a ring, a ∈ A and morphisms preserve the distinguished elements). The forgetful functor G:'''Ring'''<sub>*</sub> → '''Ring''' has a left adjoint – it assigns to every ring R the pair (R[x],x) where R[x] is the [[polynomial ring]] with coefficients from R.
* '''Abelianization'''. Consider the inclusion functor ''G'' : '''Ab''' → '''Grp''' from the [[category of abelian groups]] to [[category of groups]]. It has a left adjoint called [[abelianization]] which assigns to every group ''G'' the quotient group ''G''<sup>ab</sup>=''G''/[''G'',''G''].
* '''The Grothendieck group'''. In [[K-theory]], the point of departure is to observe that the category of [[vector bundle]]s on a [[topological space]] has a commutative monoid structure under [[Direct sum of modules|direct sum]]. One may make an [[abelian group]] out of this monoid, the [[Grothendieck group]], by formally adding an additive inverse for each bundle (or equivalence class). Alternatively one can observe that the functor that for each group takes the underlying monoid (ignoring inverses) has a left adjoint. This is a once-for-all construction, in line with the third section discussion above. That is, one can imitate the construction of [[negative number]]s; but there is the other option of an [[existence theorem]]. For the case of finitary algebraic structures, the existence by itself can be referred to [[universal algebra]], or [[model theory]]; naturally there is also a proof adapted to category theory, too.
* '''Frobenius reciprocity''' in the [[group representation|representation theory of groups]]: see [[induced representation]]. This example foreshadowed the general theory by about half a century.