# Quotient ring

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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.

Another most important example is the quotient ring , where is the ring of polynomials over in one variable and is an irreducible polynomial. This quotient ring describes all field extensions of by roots of the equation (cf. also Extension of a field).