# 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 ${\ displaystyle [X + 1] \ cdot [X + 2]}$ ${\ displaystyle [X + 1] \ cdot [X + 2] = [(X + 1) \ cdot (X + 2)] = [X ^ {2} + 3X + 2] = [3X + 1]}$ • All finite fields are obtained as factor rings of the polynomial rings over the residual class fields with prime numbers .${\ displaystyle \ mathbb {F} _ {p} = \ mathbb {Z} / p \ mathbb {Z}}$ ${\ displaystyle p}$ ## 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"