  For an associative ring we investigate a construction of Cohn's universal
ring of fractions defined relative to a multiplicative set of matrices. The
construction avoids the Ore condition, which is necessary to construct a ring
of fractions relative to a multiplicative set of elements. But a similar
condition, which we call the ``pseudo-Ore'' condition, plays an important role
in the construction of Cohn's localization. We show that this condition in fact
determines the equivalence relation used in the construction.
