Algebra Notes

[elim]

Let $R$ be a ring, $\Omega_R = \{(r,s)\mid r,s\in R,\;s\ne 0\},$ relation $\sim$ on $\Omega_R$ by
$\qquad (r,s)\sim (u,v)\iff rv = su$. Clearly $\sim$ is an equivalence relation,
$\qquad\small[r,s] = \{(u,v)\in\Omega: (u,v)\sim (r,s)\}$ as the $\sim$ equivalence class of $\small(r,s)$.

1.3.1 Proposition If $R$ is an integral domain, then $\small F_R = \{[r,s]\mid (r,s)\in\Omega_R\}$
$\qquad$forms the field of fractions of $R$ with ring operations
$\qquad [r,s]+[u,v] = [rv +su,\,sv],\;[r,s][u,v] = [ru,sv].$

$\qquad$We may identify $R$ with the subring $\{[r,1]\mid r\in R\}$ of $F_R.\;$
$\qquad$i.e. $\;i: R\to F_R\;(r\mapsto [r,1])$ is the natural monomorphism.

1.3.2 Proposition If $K$ is a field and $\phi: R\underset{\;}{\to} K$ is a monomorphism,
$\qquad$then $\phi$ extends uniquely to $F_R$, i.e. there is a unique homomorphism
$\qquad\tilde{\phi}:F_R\to K$ such that $\phi = \tilde{\phi}\circ i.$