Ring of fractions

In abstract algebra, the total quotient ring, or total ring of fractions, is a construction that generalizes the notion of the field of fractions of an integral domain to commutative rings R that may have zero divisors. The construction embeds R in a larger ring, giving every non-zero-divisor of R an inverse in the larger ring. If the homomorphism from R to the new ring is to be injective, no further elements can be given an inverse.

Let ${\displaystyle R}$ be a commutative ring and let ${\displaystyle S}$ be the set of elements which are not zero divisors in ${\displaystyle R}$; then ${\displaystyle S}$ is a multiplicatively closed set. Hence we may localize the ring ${\displaystyle R}$ at the set ${\displaystyle S}$ to obtain the total quotient ring ${\displaystyle S^{-1}R=Q(R)}$.

...
Wikipedia