Total ring of fractions
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 a domain to commutative rings that may have zero divisors. The construction embeds the ring in a larger ring, giving every non-zerodivisor of the smaller ring an inverse in the larger ring. Nothing more in the small ring can be given an inverse, because zero divisors are impossible to invert[3]. In light of this, the total ring of quotients is optimal in the sense that "everything that could have an inverse gets 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 that does not contain zero. Hence we may localize the 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 set D 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 ring R. Suppose a in R is not a zero divisor. Then in a von Neumann regular ring a=axa for some x in R, giving the equation a(xa-1)=0. Since a is not a zero divisor, xa=1, showing a is a unit. Here again, .
Applications
In algebraic geometry one considers a sheaf of total quotient rings on a scheme, and this may be used to give one possible definition of a Cartier divisor.
Generalization
If is a commutative ring and any multiplicative submagma of with unit, one can construct the in a similar fashion, where only elements of are possible denominators. If , then is the trivial ring. For details, see Localization of a ring.
Notes
- ^ Matsumura (1980), p. 12
- ^ Matsumura (1989), p. 21
- ^ If one supposes a is a nonzero zero divisor in R and also a unit in its total ring of quotients Q, then ab=0 for some nonzero b in R and ca=1 for a c in Q, and then 0=c(ab)=(ca)b=b, but b was assumed to be nonzero. This contradiction shows a zero divisor of R cannot be a unit in Q.
References
- Hideyuki Matsumura, Commutative algebra, 1980
- Hideyuki Matsumura, Commutative ring theory, 1989