S

S^{- 1} R

consists of fractions

r / s

with

r \in R, s \in S

, under the equivalence

\frac{r}{s} = \frac{r ^{'}}{s ^{'}} ⟺ \exists u \in S : u (r s^{'} - r^{'} s) = 0.

The factor

u

\frac{r}{s} + \frac{r ^{'}}{s ^{'}} = \frac{r s ^{'} + r ^{'} s}{s s ^{'}}, \frac{r}{s} \frac{r ^{'}}{s ^{'}} = \frac{r r ^{'}}{s s ^{'}} .

r

r /1

and makes every element of

S

a unit. It is universal with this property: a ring homomorphism

ϕ : R \to T

taking

S

into units extends uniquely by

r / s \mapsto ϕ (r) ϕ (s)^{- 1}

. Usually zero is excluded from

S

; if it is included, this construction is the zero ring.

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