Editing Formal power series (section)

===Replacing the index set by an ordered abelian group===
{{Main|Hahn series}}

Suppose <math>G</math> is an ordered [[abelian group]], meaning an abelian group with a total ordering <math><</math> respecting the group's addition, so that <math>a<b</math> if and only if <math>a+c<b+c</math> for all <math>c</math>. Let '''I''' be a [[well-order]]ed subset of <math>G</math>, meaning '''I''' contains no infinite descending chain. Consider the set consisting of

:<math>\sum_{i \in I} a_i X^i </math>

for all such '''I''', with <math>a_i</math> in a commutative ring <math>R</math>, where we assume that for any index set, if all of the <math>a_i</math> are zero then the sum is zero. Then <math>R((G))</math> is the ring of formal power series on <math>G</math>; because of the condition that the indexing set be well-ordered the product is well-defined, and we of course assume that two elements which differ by zero are the same. Sometimes the notation <math>[[R^G]]</math> is used to denote <math>R((G))</math>.<ref>{{cite journal | first1=Khodr | last1=Shamseddine | first2=Martin | last2=Berz | url= http://www.physics.umanitoba.ca/~khodr/Publications/RS-Overview-offprints.pdf | title=Analysis on the Levi-Civita Field: A Brief Overview | journal= Contemporary Mathematics | volume=508 | pages=215–237 | date=2010| doi=10.1090/conm/508/10002 | isbn=9780821847404 }}</ref>

Various properties of <math>R</math> transfer to <math>R((G))</math>. If <math>R</math> is a field, then so is <math>R((G))</math>. If <math>R</math> is an [[ordered field]], we can order <math>R((G))</math> by setting any element to have the same sign as its leading coefficient, defined as the least element of the index set '''I''' associated to a non-zero coefficient. Finally if <math>G</math> is a [[divisible group]] and <math>R</math> is a [[real closed field]], then <math>R((G))</math> is a real closed field, and if <math>R</math> is [[algebraically closed]], then so is <math>R((G))</math>.

This theory is due to [[Hans Hahn (mathematician)|Hans Hahn]], who also showed that one obtains subfields when the number of (non-zero) terms is bounded by some fixed infinite cardinality.