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 ringsR 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.
Definition
Let be a commutative ring and let be the set of elements which are not zero divisors in ; then is a multiplicatively closed set. Hence we may localizethe ring at the set to obtain the total quotient ring. If is a domain, then and the total quotient ring is the same as the field of fractions. This justifies the notation, which is sometimes used for the field of fractions as well, since there is no ambiguity in the case of a domain. Since in the construction contains no zero divisors, the natural map is injective, so the total quotient ring is an extension of.
Examples
The total quotient ring of a product ring is the product of total quotient rings. In particular, if A and B are integral domains, it is the product of quotient fields. The total quotient ring of the ring of holomorphic functions on an open setD of complex numbers is the ring of meromorphic functions on D, even if D is not connected. In an Artinian ring, all elements are units or zero divisors. Hence the set of non-zero divisors is the group of units of the ring,, and so. But since all these elements already have inverses,. The same thing happens in a commutative von Neumann regular ringR. Suppose a in R is not a zero divisor. Then in a von Neumann regular ringa = axa for some x in R, giving the equation a = 0. Since a is not a zero divisor, xa = 1, showing a is a unit. Here again,.
There is an important fact: Proof: Every element ofQ is either a unit or a zerodivisor. Thus, any proper idealI of Q must consist of zerodivisors. Since the set of zerodivisors of Q is the union of the minimal prime ideals as Q is reduced, by prime avoidance, I must be contained in some. Hence, the ideals are the maximal ideals of Q, whose intersection is zero. Thus, by the Chinese remainder theorem applied to Q, we have: Finally, is the residue field of. Indeed, writing S for the multiplicatively closed set of non-zerodivisors, by the exactness of localization, which is already a field and so must be.
Generalization
If is a commutative ring and is any multiplicative subset in, the localization can still be constructed, but the ring homomorphism from to might fail to be injective. For example, if, then is the trivial ring.
