Total quotient ring

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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. Nothing more in A can be given an inverse, if one wants the homomorphism from A to the new ring to be injective.

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 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

  • The rational functions over a ring R{{ safesubst:#invoke:Unsubst||$N=Dubious |date=__DATE__ |$B=

{{#invoke:Category handler|main}}[dubious ] }} can be constructed from the polynomial ring R[x] as a total quotient ring.[3]

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.

Notes

  1. Matsumura (1980), p. 12
  2. Matsumura (1989), p. 21
  3. {{#invoke:citation/CS1|citation |CitationClass=citation }}.

References

  • Hideyuki Matsumura, Commutative algebra, 1980
  • Hideyuki Matsumura, Commutative ring theory, 1989

de:Lokalisierung_(Algebra)#Totalquotientenring