- 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 aprincipal 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 ofcyclic code s over a finite field "F", which are ideals of "F" ["X"] ⁄ ("X"^{"n"}− 1).**Examples*** The ring

**Z**ofinteger s 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 ofGaussian integers ,**Z**["i"] , form a principal ideal ring;

* theEisenstein integers ,**Z**["ω"] , where "ω" is a cube root of 1, form a principal ideal ring.* The polynomial ring

**Z**[√5] =**Z**⊕ √5**Z**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.*

### 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**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**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