Editing Field of fractions (section)

{{Short description|Abstract algebra concept}}
{{redirect-distinguish|Quotient field|Quotient ring}}
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In [[abstract algebra]], the '''field of fractions''' of an [[integral domain]] is the smallest [[field (mathematics)|field]] in which it can be [[Embedding|embedded]]. The construction of the field of fractions is modeled on the relationship between the integral domain of [[integer]]s and the field of [[rational number]]s. Intuitively, it consists of ratios between integral domain elements.

The field of fractions of an integral domain <math>R</math> is sometimes denoted by <math>\operatorname{Frac}(R)</math> or <math>\operatorname{Quot}(R)</math>, and the construction is sometimes also called the '''fraction field''', '''field of quotients''', or '''quotient field''' of <math>R</math>.  All four are in common usage, but are not to be confused with the [[Quotient ring|quotient of a ring by an ideal]], which is a quite different concept. For a [[commutative ring]] that is not an integral domain, the analogous construction is called the [[Localization (commutative algebra)|localization]] or ring of quotients.