# Factor ring

In the Algebra refers to a particular type of rings than **factor ring** or *quotient ring* or *residue class ring* . It is a generalization of the residue class rings of integers .

## definition

If a ring is and a (both-sided) ideal of , then the set of equivalence classes modulo forms a ring with the following connections:

where is defined as .

This ring is called the factor ring modulo or remainder class ring or quotient ring. (However, it has nothing to do with the terms quotient field or total quotient ring ; these are localizations .)

## Examples

- The set of all integer multiples of is an ideal in , and the factor ring is the remainder class ring modulo .

- If a polynomial is over a commutative unitary ring , then the set of all polynomial multiples of is an ideal in the polynomial ring , and the factor ring is modulo .

- If we consider the polynomial over the field of real numbers , the factor ring is isomorphic to the field of complex numbers ; the equivalence class of corresponds to the imaginary unit .

- Sample calculations:
- Because of this, the polynomial is in the same equivalence class modulo as .
- We determine for the product

- All finite fields are obtained as factor rings of the polynomial rings over the residual class fields with prime numbers .

## properties

- If there is a commutative ring with one element, then an ideal is a prime ideal if and only if there is an integrity ring .
- If a commutative ring is one element, then an ideal is a maximum ideal if and only if there is a body.
- If a field and an irreducible polynomial are over , then there is a maximal ideal in and therefore is a field. This body is an upper body of , in which has a zero (the remainder class of ). The body expansion is finite and algebraic , its degree coincides with the degree of . If one repeats the procedure with the non-linear irreducible divisors of , one finally obtains a field in which is divided into linear factors : The decay field of .

## Ideal theory

Be a commutative ring with one element and an ideal. Than are

- the ideals of the ring exactly the ideals of that contain (thus )
- the prime ideals of the ring exactly the prime ideals of that contain
- the maximum ideals of the ring contain exactly the maximum ideals of that

## comment

The term is to be distinguished from the factorial ring , in which the unique prime factorization exists.

## literature

- Kurt Meyberg,
*Algebra I*, Carl Hanser Verlag (1980), ISBN 3-446-13079-9 , Chapter 3: "Rings"