@MISC{IDEAL_2.MIZ,
  AUTHOR = {Watase, Yasushige},
  TITLE = {On Primary Ideals. {P}art {I}},
  SECTION1 = {Preliminaries: Operations on Ideals},
  SECTION2 = {Definition of Primary Ideal and p-Primary},
  SECTION3 = {Some Prositions on Primary Ideals},
  DAY = {30},
  MONTH = {June},
  YEAR = {2021},
  ADDRESS1 = {Suginami-ku Matsunoki 3-21-6 Tokyo\\Japan},
  SUMMARY = {We formalize in the Mizar System \cite{Mizar-State-2015},
   definitions and basic propositions about primary ideals of a
   commutative ring along with Chapter 4 of \cite{AM}.
   Additionally other necessary basic ideal operations such as
   compatibilities taking radical and intersection of finite number
   of ideals are formalized as well in order to prove theorems
   relating primary ideals.
   These basic operations are mainly quoted from Chapter 1
   of \cite{AM} and compiled as preliminaries in the
   first half of the article.},
  MSC2010 = {13A70,16D70},
  KEYWORDS = {Primary Ideal; Radical Ideal; Prime Ideal}}

@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},
  series ={Cambridge Studies in Advanced Mathematics},
  year={1989},
  edition = {2nd},
  publisher={Cambridge University Press}}
