Let R be a ring and I be a (two-sided) ideal. The quotient ring (or factor ring) R/I is the ring which has underlying additive group the quotient group R/I but with multiplication defined by (a+I)(b+I)=(ab+I). This is well defined because I is an ideal. (Note that we write the cosets additively.)

There is a canonical ring homomorphism p:R->R/I defined by p(a)=a+I.

See also isomorphism theorems.