Principal ideal ring

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In mathematics, a principal ideal ring is a ring R such that every ideal I of R is a principal ideal, i.e. generated by a single element a of R.

A principal ideal ring which is also an integral domain is said to be a principal ideal domain (PID).

Every quotient ring of a principal ideal ring is again a principal ideal ring. This has application to the study of cyclic codes over a finite field F, which are ideals of F[X] ⁄ (Xn − 1).

[edit] Examples

  • The ring Z of integers with the usual operations is a principal ideal ring;
  • F[X], the ring of polynomials in one variable X with coefficients in a field F, is a principal ideal ring;
  • the ring of Gaussian integers, Z[i], form a principal ideal ring;
  • the Eisenstein integers, Z[ω], where ω is a cube root of 1, form a principal ideal ring.
  • The polynomial ring Z[√5] = Z ⊕ √5Z is not a principal ideal ring: there is no single element r ∈ Z[√5] such that the ideal generated by r equals the ideal generated by the two elements 2 and √5.

[edit] References

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