@MISC{RINGFRAC.MIZ,
  AUTHOR = {Watase, Yasushige},
  TITLE = {{R}ings of {F}ractions and {L}ocalization},
  SECTION1 = {Preliminaries: Units, Zero Divisors, multiplicatively-closed Set},
  SECTION2 = {Equivalence Relation of Fractions},
  SECTION3 = {Constriction of Ring of Fractions},
  SECTION4 = {Localization in terms of Prime Ideals},
  SECTION5 = {Universal Property of Ring of Fractions},
  SECTION6 = {The total-quotient ring and the Quotient
    Field of Integral Domain},
  DAY = {13},
  MONTH = {January},
  YEAR = {2020},
  ADDRESS1 = {Suginami-ku Matsunoki 3-21-6 Tokyo\\Japan},
 SUMMARY = {This article formalized rings of fractions in Mizar System
 \cite{Mizar-State-2015}.
  A construction of the ring of fractions from an integral domain,
  namely a quotient field was formalized in \cite{QUOFIELD}. \par
  This article generalized the construction of
    fractions and enabled to apply to an arbitrary commutative ring
    by means of an equivalence relation which is based on a multiplicatively
    closed set which defines a fraction. \par
  Localization of rings, appeared in Commutative Ring theory
  or in Algebraic Geometry, is a specific ring of fractions in terms of
  a prime ideals and a localized ring turns to be a local ring. \par
  This article also provided formal proof of a universal property
  of a ring of fractions, the total-quotient ring, a proof of
  the equivalence between the total-quotient ring and the quotient field
  of an integral domain.},
  MSC2010 = {13B30,16S85},
  KEYWORDS = {rings of fractions, localization, total-quotient
  ring, quotient field}}

@BOOK{AM,
 AUTHOR={M.F.Atiyah and I.G. MacDonald},
 TITLE={Introduction to Commutative Algebra},
 PUBLISHER={Addison-Wesley Publishing Company, Inc.},
 YEAR=1969}

@book{matsumura1989,
  title={Commutative Ring Theory},
  author={Matsumura, Hideyuki},
  note ={Cambridge Studies in Advanced Mathematics},
  year={1989},
  edition = {2nd},
  publisher={Cambridge University Press}}
