Ring homomorphisms Algebraic structures Related structures Algebraic number theory Noncommutative algebraic geometry Free algebra Clifford algebra In abstract algebra, the total quotient ring[1] or total ring of fractions[2] 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.
be a commutative ring and let
be the set of elements that are not zero divisors in
is a multiplicatively closed set.
Hence we may localize the ring
to obtain the total quotient ring
and the total quotient ring is the same as the field of fractions.
, which is sometimes used for the field of fractions as well, since there is no ambiguity in the case of a domain.
in the construction contains no zero divisors, the natural map
is injective, so the total quotient ring is an extension of
Proposition — Let A be a reduced ring that has only finitely many minimal prime ideals,
(e.g., a Noetherian reduced ring).
is the Artinian scheme consisting (as a finite set) of the generic points of the irreducible components of
Proof: Every element of Q(A) is either a unit or a zero divisor.
Thus, any proper ideal I of Q(A) is contained in the set of zero divisors of Q(A); that set equals the union of the minimal prime ideals
By prime avoidance, I must be contained in some
Thus, by the Chinese remainder theorem applied to Q(A), Let S be the multiplicatively closed set of non-zero-divisors of A.
By exactness of localization, which is already a field and so must be
is a commutative ring and
is any multiplicatively closed set in
can still be constructed, but the ring homomorphism from