Homomorphisms and Ideals

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


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.


Prime Ideal Finite Field Maximal Ideal Commutative Ring Residue Class 
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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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