Quotient Ring

algebra

Definition

Quotient Ring

Let be a ring and let be an ideal, that is, an additive subgroup of that is closed under multiplication by elements of . The quotient ring of modulo is the set of cosets

Addition and multiplication are defined by

These operations are well-defined because is an ideal.

Example

For the polynomial ring and the ideal ,

This identifies the residue class of with the imaginary unit .