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 .
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 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 .
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"