# 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: ${\ displaystyle (R, +, \ cdot)}$${\ displaystyle I}$${\ displaystyle R}$${\ displaystyle R / I = \ left \ {a + I \ mid a \ in R \ right \}}$ ${\ displaystyle I}$

• ${\ displaystyle (a + I) + (b + I): = (a + b) + I}$
• ${\ displaystyle (a + I) \ cdot (b + I): = a \ cdot b + I,}$

where is defined as . ${\ displaystyle (a + I)}$${\ displaystyle \ {a + r \, | \, r \ in I \}}$

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 .) ${\ displaystyle R}$${\ displaystyle I}$

## Examples

• The set of all integer multiples of is an ideal in , and the factor ring is the remainder class ring modulo .${\ displaystyle n \ mathbb {Z}}$${\ displaystyle n}$${\ displaystyle \ mathbb {Z}}$${\ displaystyle \ mathbb {Z} / n \ mathbb {Z}}$${\ displaystyle n}$
• 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 .${\ displaystyle f \ in R [X]}$ ${\ displaystyle R}$${\ displaystyle R [X] \ cdot f = (f)}$${\ displaystyle f}$ ${\ displaystyle R [X]}$${\ displaystyle R [X] / (f) = \ left \ {g + (f) \ mid g \ in R [X] \ right \}}$${\ displaystyle R [X]}$${\ displaystyle f}$
• If we consider the polynomial over the field of real numbers , the factor ring is isomorphic to the field of complex numbers ; the equivalence class of corresponds to the imaginary unit .${\ displaystyle f = X ^ {2} +1}$${\ displaystyle \ mathbb {R}}$${\ displaystyle \ mathbb {R} [X] / (f)}$${\ displaystyle X}$${\ displaystyle \ mathrm {i}}$
Sample calculations:
Because of this, the polynomial is in the same equivalence class modulo as .${\ displaystyle X ^ {2}}$${\ displaystyle X ^ {2} = f-1}$${\ displaystyle f}$${\ displaystyle -1}$
We determine for the product ${\ displaystyle [X + 1] \ cdot [X + 2]}$${\ displaystyle [X + 1] \ cdot [X + 2] = [(X + 1) \ cdot (X + 2)] = [X ^ {2} + 3X + 2] = [3X + 1]}$

## 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 .${\ displaystyle R}$${\ displaystyle I}$${\ displaystyle R / I}$
• If a commutative ring is one element, then an ideal is a maximum ideal if and only if there is a body.${\ displaystyle R}$${\ displaystyle I}$${\ displaystyle R / I}$
• 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 .${\ displaystyle K}$${\ displaystyle f}$${\ displaystyle K}$${\ displaystyle (f)}$${\ displaystyle K [X]}$${\ displaystyle L \ colon = K [X] / (f)}$${\ displaystyle K}$${\ displaystyle f}$${\ displaystyle X}$ ${\ displaystyle L / K}$${\ displaystyle f}$${\ displaystyle L}$${\ displaystyle f}$${\ displaystyle f}$${\ displaystyle f}$

## Ideal theory

Be a commutative ring with one element and an ideal. Than are ${\ displaystyle R}$${\ displaystyle I \ subseteq R}$

• the ideals of the ring exactly the ideals of that contain (thus )${\ displaystyle R / I}$${\ displaystyle J}$${\ displaystyle R}$${\ displaystyle I}$${\ displaystyle I \ subseteq J}$
• the prime ideals of the ring exactly the prime ideals of that contain${\ displaystyle R / I}$${\ displaystyle R}$${\ displaystyle I}$
• the maximum ideals of the ring contain exactly the maximum ideals of that${\ displaystyle R / I}$${\ displaystyle R}$${\ displaystyle I}$

## 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"