Exercise 15, § 9

Let A be a non-commutative ring with no divisors of 0. A is said to admit a field of left fractions if it is isomorphic to a subring B of a field K such that every element of K is of the form $x^{-1}y$, where $x \in B, y \in B$.

(a) Let $A'$ be the set of elements $\neq 0$ in A. For A to admit a field of left fractions, it is necessary that the following condition be fulfilled: (G) for all $x \in A, x' \in A'$, there exist $u \in A'$ and $v \in A$ such that $ux = vx'$. (b) Suppose conversely that condition (G) is fulfilled. Show that in the set $A \times A'$ the relation R between $(x, x')$ and $(y, y')$ which states "for every ordered pair $(u, v)$ of elements $\neq 0$ such that $ux' = vy', ux = vy"$ is an equivalence relation.

Let $(x, x'), (y, y')$ be two elements of $A \times A'$, $\xi$ and $\eta$ their respective classes (mod. R). For every ordered pair $(u, u') \in A \times A'$ such that $u'x = uy'$, show that the class (mod. R) of $(uy, u'y')$ depends only on the classes $\xi$ and $\eta$; if it is denoted by $\xi \eta$, a law of composition is defined on the set $K = (A \times A')/R$. If $K'$ is the set of elements of K distinct from the class 0 of elements $(0, x')$ of $A \times A'$, $K'$, with the law induced by the above law, is a group.

For every element $x \in A$, the elements $(x'x, x')$, where $x'$ runs through $A'$, define an isomorphism of A (with multiplication alone) onto a subring of K. Identifying A with its image under this isomorphism, the class (mod. R) of an ordered pair $(x, x') \in A \times A'$ is identified with the element ${x'}^{-1}x$.

This being so, if $\xi = {x'}^{-1}x$ is an element of K and 1 denotes the unit element of $K'$, then $\xi + 1$ denotes the element ${x'}^{-1}(x + x')$, which does not depend on the representation of $\xi$ in the form ${x'}^{-1}x$. We then write $\xi + 0 = \xi$ and, for $\eta \neq 0$, $\xi + \eta = \eta(\eta^{-1}\xi + 1)$. Show that the addition and multiplication thus defined on K determine on that set a field structure which extends the ring structure on A; in other words, condition (G) is sufficient for A to admit a field of left fractions.