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Homomorphisms and Ideals

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

Abstract

A further difference of principle between arbitrary commutative rings and fields is the existence of nontrivial homomorphisms. A homomorphism of a ring A to a ring B is a map f: AB such that
$$ f({a_1} + {a_2}) = f({a_1}) + f({a_2}),{\mkern 1mu} f({a_1})\cdot f({a_2}){\mkern 1mu} and{\mkern 1mu} f({1_A}) = {1_B} $$
(we write 1 A and 1 B for the identity elements of A of B). An isomorphism is a homomorphism having an inverse.

Keywords

Prime Ideal Finite Field Maximal Ideal Commutative Ring Residue Class 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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