# Quotient ring

of a ring by an ideal

The quotient group of the additive group of by the subgroup , with multiplication

The quotient turns out to be a ring and is denoted by . The mapping , where , is a surjective ring homomorphism, called the natural homomorphism (see Algebraic system).

The most important example of a quotient ring is the ring of residues modulo — the quotient ring of the ring of integers by the ideal . The elements of can be assumed to be the numbers , where the sum and the product are defined as the remainders on diving the usual sum and product by . One can establish a one-to-one order-preserving correspondence between the ideals of and the ideals of containing . In particular, is simple (cf. Simple ring) if and only if is a maximal ideal.