Editing Field of fractions (section)

=== Semifield of fractions ===
The '''semifield of fractions''' of a [[commutative semiring]] in which every nonzero element is (multiplicatively) cancellative is the smallest [[semifield]] in which it can be [[Embedding|embedded]]. (Note that, unlike the case of rings, a semiring with no [[zero divisor]]s can still have nonzero elements that are not cancellative. For example, let <math>\mathbb{T}</math> denote the [[tropical semiring]] and let <math>R=\mathbb{T}[X]</math> be the [[polynomial ring#polynomial rigs|polynomial semiring]] over <math>\mathbb{T}</math>. Then <math>R</math> has no zero divisors, but the element <math>1+X</math> is not cancellative because <math>(1+X)(1+X+X^2)=1+X+X^2+X^3=(1+X)(1+X^2)</math>).

The elements of the semifield of fractions of the commutative [[semiring]] <math>R</math> are [[equivalence class]]es written as
:<math>\frac{a}{b}</math>
with <math>a</math> and <math>b</math> in <math>R</math> and <math>b\neq 0</math>.