Principal ideal ring

Principal ideal ring

In mathematics, a principal ideal ring, or simply principal 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"] ⁄ ("X""n" − 1).

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.

References

* S. Lang, "Algebra (3 ed)", Addison-Wesley, 1993, ISBN 0-201-55540-9. Pp.86, 146-155.


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