# principal ideal

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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. 2012.

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

• 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 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… …   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 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

• principal ideal domain — noun An integral domain in which every ideal is a principal ideal …   Wiktionary

• Structure theorem for finitely generated modules over a principal ideal domain — In mathematics, in the field of abstract algebra, the structure theorem for finitely generated modules over a principal ideal domain is a generalization of the fundamental theorem of finitely generated abelian groups and roughly states that… …   Wikipedia