Localization of Ring Exists/Lemma 2

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Lemma

The operations $+$ and $\cdot$ are well defined on $A_S$.



Proof

Let:

$a / s = c / u$
$b / t = d / v$

be two sets of representatives for two distinct equivalence classes in $A_S$.

We have $w, z \in S$ such that:

$\left({a u - c s}\right) w = 0$

and

$\left({b v - d t}\right) z = 0$

Therefore:

$z w \left[{\left({a u - c s}\right) w - \left({b v - d t}\right) z}\right] = 0$

and:

$z w \left[{\left({a t + b s}\right) u v - \left({c v + d u}\right)s t}\right]$

So:

$\left({a t + b s, s t}\right) \sim \left({c v + d u, u v}\right)$

That is:

$\dfrac {a t + b s} {s t} = \dfrac {c v + d u} {u v}$


For multiplication, with $z, w$ as above we have:

$\left({a b u v - c d s t}\right)z w = 0$

So:

$\left({a b, s t}\right) \sim \left({d c, u v}\right)$

and:

$\dfrac {a b} {s t} = \dfrac {d c} {u v}$

$\blacksquare$