field of quotients

English

Noun

field of quotients (plural fields of quotients)

1. (algebra) A field all of whose elements can be represented as ordered pairs each of whose components belong to a given integral domain, such that the second component is non-zero, and so that the additive operator is defined like so: $(a,b)+(a',b')=(ab'+a'b,bb')$, the multiplicative operator is defined coordinate-wise, the zero is $(0,1)$, the unity is $(1,1)$, the additive inverse of $(a,b)$ is $(-a,b)$, equivalence is defined like so: $(a,b)\equiv (a',b')$ if and only if $ab'=a'b$, and multiplicative inverse of a non-zero–equivalent element $(a,b)$ is $(b,a)$.

Synonyms

• field of fractions

See also

• localization