## Abstract

Let I be an ideal in in particular, the identity element is 0 +

*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*r*∈*R*, and addition is given by$$ \left( {r{ + }I} \right){ + }\left( {r' + I} \right) = \left( {r + r'} \right) + I $$

*I*=*I.*Recall that*r*+*I*=*r′*+*I*if and only if*r-r′*∈*I*. Finally, remember that the**natural map***π*:*R*→*R/I*is the (group) homomorphism defined by*r*↦*r*+*I*.## Copyright information

© Springer-Verlag New York Inc. 1990