Encyclopedia > Principal ideal domain

  Article Content

Principal ideal domain

In abstract algebra, a principal ideal domain (PID) is an integral domain in which every ideal is principal (that is, generated by a single element).

Examples are the ring of integers, all fields, and rings of polynomials in one variable with coefficients in a field. All euclidean domains are principal ideal domains. The ring Z[X] of all polynomials with integer coefficients is not principal, since for example the ideal generated by 2 and X cannot be generated by a single polynomial.

In a principal ideal domain, any two elements have a greatest common divisor (and may have more than one).

In all rings, maximal ideals are prime. In principal ideal domains a near converse holds: every nonzero prime ideal is maximal.

Every principal ideal domain is Noetherian and a unique factorization domain.

All Wikipedia text is available under the terms of the GNU Free Documentation License

  Search Encyclopedia

Search over one million articles, find something about almost anything!
  Featured Article
David McReynolds

... Today, McReynolds lives in New York City and continues to be active in the Socialist and pacifist movements through the combination of Marxian politics and Gandhia ...

This page was created in 74.5 ms