Quotient ring

In algebra refers to a specific type of rings as a factor ring or quotient ring or residue class ring. This is a generalization of the residue class rings of integers.

Definition

Is a ring and a (two-sided ) ideal of, then is the set of equivalence classes modulo a ring with the following links:

This ring is called the factor ring modulo or residue class ring or ring of quotients. ( 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 residue class ring modulo.
  • If a polynomial over an integral domain, then the set of all polynomial multiples of an ideal in a polynomial ring, and is the factor ring modulo.
  • Consider the polynomial over the field of real numbers, so the factor ring is isomorphic to the field of complex numbers; the equivalence class of case corresponds to the imaginary unit.
  • You receive all the finite field as a factor rings of polynomial rings over the residue class bodies.

Properties

  • An ideal I R is a commutative ring with identity, then is a prime ideal if and only if an integrity ring.
  • An ideal I R is a commutative ring with identity, then is a maximal ideal if and only if a body is.
  • If K is a field and f is an irreducible polynomial over K, then a maximal ideal in and is therefore a body. This body is an upper body of K, where f is a zero ( the coset of X). The field extension is finite and algebraic, its degree coincides with the degree of f coincide.

If you repeat the process with the L on non-linear irreducible divisors of f, is finally obtained a body decomposes in the f into linear factors: the splitting field of f

Ideal theory

Be a commutative ring with 1 and an ideal. Then

  • The ideals of the ring exactly the ideals of that contain (ie )
  • The prime ideals of the ring exactly the prime ideals of containing
  • The maximum ideals of the ring exactly the maximal ideals of, containing

Remark

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

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