Commutative Rings

  • Igor R. Shafarevich
Part of the Encyclopaedia of Mathematical Sciences book series (EMS, volume 11)


The simplest possible example of ‘coordinatisation’ is counting, and it leads (once 0 and negative numbers have been introduced) to the integers, which do not form a field. Operations of addition and multiplication are defined on the set of all integers (positive, zero or negative), and these satisfy all the field axioms but one, namely the existence of an inverse element a −1 for every a ≠ 0 (since, for example, ½ is already not an integer).


Commutative Ring Integral Domain Polynomial Ring Formal Power Series Laurent Series 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Igor R. Shafarevich
    • 1
  1. 1.Steklov Mathematical InstituteRussian Academy of ScienceMoscowRussia

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