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"