quotient ring

A quotient ring is what you get when you decide to treat all the elements of an ideal as if they were zero — you 'mod out' by the ideal and study the simplified arithmetic that remains. It is the ring-theoretic analogue of forming a quotient group, and it is the universal way to impose relations on a ring.

Given a ring R and a two-sided ideal I, the quotient ring R/I consists of the cosets a + I, with addition (a + I) + (b + I) = (a + b) + I and multiplication (a + I)(b + I) = a·b + I. The ideal condition on I is exactly what makes multiplication well-defined on cosets. The natural projection R → R/I, sending a to a + I, is a surjective ring homomorphism with kernel I.

Quotient rings are governed by the first isomorphism theorem: for any ring homomorphism f from R, the image is isomorphic to R/ker(f). The character of the quotient encodes the character of the ideal — R/P is an integral domain precisely when P is prime, and R/M is a field precisely when M is maximal. This dictionary turns questions about ideals into questions about quotient rings and back.