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.