Quotient Rings

  • Joseph Rotman
Part of the Universitext book series (UTX)


Let I be an ideal in R. Forgetting the multiplication for a moment, I is a subgroup of the additive group of R; moreover, R abelian implies that I is a normal subgroup, and so the quotient group R/I exists. The elements of R/I are the cosets r + I, where rR, and addition is given by
$$ \left( {r{ + }I} \right){ + }\left( {r' + I} \right) = \left( {r + r'} \right) + I $$
in particular, the identity element is 0 + I = I. Recall that r + I = r′ + I if and only if r-r′I. Finally, remember that the natural map π: RR/I is the (group) homomorphism defined by rr + I.

Copyright information

© Springer-Verlag New York Inc. 1990

Authors and Affiliations

  • Joseph Rotman
    • 1
  1. 1.Department of MathematicsUniversity of Illinois at Urbana-ChampaignUrbanaUSA

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