Total ring of fractions

From Wikipedia, the free encyclopedia
  (Redirected from Total quotient ring)
Jump to: navigation, search

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[edit]

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[edit]

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, .

The total ring of fractions of a reduced ring[edit]

There is an important fact:

Proposition — Let A be a Noetherian reduced ring with the minimal prime ideals . Then

Geometrically, 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 zerodivisor. Thus, any proper ideal I of Q(A) must consist of zerodivisors. Since the set of zerodivisors of Q(A) is the union of the minimal prime ideals as Q(A) is reduced, by prime avoidance, I must be contained in some . Hence, the ideals are the maximal ideals of Q(A), whose intersection is zero. Thus, by the Chinese remainder theorem applied to Q(A), 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[edit]

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[edit]

  1. ^ Matsumura (1980), p. 12
  2. ^ Matsumura (1989), p. 21

References[edit]

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