Question

proof ring of fraction step by step (in general)

Answer 1

Ring of Fractions:

The term "ring of fractions" is sometimes used to denote any localization of a ring. The ring of fractions in the above meaning is then referred to as the total ring of fractions, and coincides with the localization with respect to the set of all non-zero divisors.The extension ring obtained from a commutative unit ring (other than the trivial ring) when allowing division by all non-zero divisors. The ring of fractions of an integral domain is always a field.

When defining addition and multiplication of fractions, all that is required of the denominators is that they be multiplicatively closed, i.e., if , then ,

			(1)
			(2)

Given a multiplicatively closed set in a ring , the ring of fractions is all elements of the form with and . Of course, it is required that and that fractions of the form and be considered equivalent. With the above definitions of addition and multiplication, this set forms a ring.

The original ring may not embed in this ring of fractions if it is not an integral domain. For instance, if for some , then in the ring of fractions.When the complement of is an ideal , it must be a prime ideal because is multiplicatively closed. In this case, the ring of fractions is the localization at .

When the ring is an integral domain, then the nonzero elements are multiplicatively closed. Letting be the nonzero elements, then the ring of fractions is a field called the field of fractions, or the total ring of fractions. In this case one can also use the usual rule for division of fractions, which is not normally available for more general .

The Total Ring of Fractions of a Reduced Ring:

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

Geometricallyis 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