Editing Ring (mathematics) (section)

=== Limits and colimits of rings ===
Let {{mvar|R{{sub|i}}}} be a sequence of rings such that {{mvar|R{{sub|i}}}} is a subring of {{math|''R''{{sub|''i'' + 1}}}} for all {{mvar|i}}. Then the union (or [[filtered colimit]]) of {{mvar|R{{sub|i}}}} is the ring <math>\varinjlim R_i</math> defined as follows: it is the disjoint union of all {{mvar|R{{sub|i}}}}'s modulo the equivalence relation {{math|''x'' ~ ''y''}} if and only if {{math|1=''x'' = ''y''}} in {{mvar|R{{sub|i}}}} for sufficiently large {{mvar|i}}.

Examples of colimits:
* A polynomial ring in infinitely many variables: <math>R[t_1, t_2, \cdots] = \varinjlim R[t_1, t_2, \cdots, t_m].</math>
* The [[algebraic closure]] of [[finite field]]s of the same characteristic <math>\overline{\mathbf{F}}_p = \varinjlim \mathbf{F}_{p^m}.</math>
* The field of [[formal Laurent series]] over a field {{mvar|k}}: <math>k(\!(t)\!) = \varinjlim t^{-m}k[\![t]\!]</math> (it is the field of fractions of the [[formal power series ring]] <math>k[\![t]\!].</math>)
* The [[function field of an algebraic variety]] over a field {{mvar|k}} is <math>\varinjlim k[U]</math> where the limit runs over all the coordinate rings {{math|''k''[''U'']}} of nonempty open subsets {{mvar|U}} (more succinctly it is the [[stalk (mathematics)|stalk]] of the structure sheaf at the [[generic point]].)

Any commutative ring is the colimit of [[finitely generated ring|finitely generated subrings]].<!-- non-commutative case? -->

A [[projective limit]] (or a [[filtered limit]]) of rings is defined as follows. Suppose we are given a family of rings {{math|''R''{{sub|''i''}}}}, {{math|''i''}} running over positive integers, say, and ring homomorphisms {{math|''R''{{sub|''j''}} → ''R''{{sub|''i''}}}}, {{math|''j'' ≥ ''i''}} such that {{math|''R''{{sub|''i''}} → ''R''{{sub|''i''}}}} are all the identities and {{math|''R''{{sub|''k''}} → ''R''{{sub|''j''}} → ''R''{{sub|''i''}}}} is {{math|''R''{{sub|''k''}} → ''R''{{sub|''i''}}}} whenever {{math|''k'' ≥ ''j'' ≥ ''i''}}. Then <math>\varprojlim R_i</math> is the subring of <math>\textstyle \prod R_i</math> consisting of {{math|(''x''{{sub|''n''}})}} such that {{math|''x''{{sub|''j''}}}} maps to {{math|''x''{{sub|''i''}}}} under {{math|''R''{{sub|''j''}} → ''R''{{sub|''i''}}}}, {{math|''j'' ≥ ''i''}}.

For an example of a projective limit, see ''{{slink||Completion}}''.