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.


Wikimedia Foundation. 2010.

Игры ⚽ Поможем решить контрольную работу

Look at other dictionaries:

  • Principal ideal — In ring theory, a branch of abstract algebra, a principal ideal is an ideal I in a ring R that is generated by a single element a of R .More specifically: * a left principal ideal of R is a subset of R of the form R a := { r a : r in R }; * a… …   Wikipedia

  • principal ideal domain — Math. a commutative integral domain with multiplicative identity in which every ideal is principal. Also called principal ideal ring. [1960 65] * * * …   Universalium

  • principal ideal domain — Math. a commutative integral domain with multiplicative identity in which every ideal is principal. Also called principal ideal ring. [1960 65] …   Useful english dictionary

  • Ideal (ring theory) — In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept allows the generalization in an appropriate way of some important properties of integers like even number or multiple of 3 . For instance, in… …   Wikipedia

  • Principal ideal domain — In abstract algebra, a principal ideal domain, or PID is an integral domain in which every ideal is principal, i.e., can be generated by a single element.Principal ideal domains are thus mathematical objects which behave somewhat like the… …   Wikipedia

  • Principal ideal theorem — This article is about the Hauptidealsatz of class field theory. You may be seeking Krull s principal ideal theorem, also known as Krull s Hauptidealsatz, in commutative algebra In mathematics, the principal ideal theorem of class field theory, a… …   Wikipedia

  • Free ideal ring — In mathematics, especially in the field of ring theory, a (left) free ideal ring, or fir, is a ring in which all left ideals are free of unique rank. A ring such that all left ideals with at most n generators is free of unique rank is called an n …   Wikipedia

  • principal ideal — Math. the smallest ideal containing a given element in a ring; an ideal in a ring with a multiplicative identity, obtained by multiplying each element of the ring by one specified element. [1935 40] * * * …   Universalium

  • principal ideal — Math. the smallest ideal containing a given element in a ring; an ideal in a ring with a multiplicative identity, obtained by multiplying each element of the ring by one specified element. [1935 40] …   Useful english dictionary

  • principal ideal — noun An ideal which is generated by a single element of the ring …   Wiktionary

Share the article and excerpts

Direct link
… Do a right-click on the link above
and select “Copy Link”