- Principal ideal
In

ring theory , a branch ofabstract 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 asubset of "R" of the form "R"*a*:= {"r"*a*: "r" in "R"};

* a "right principal ideal" is a subset of the form "a"*R*:= {"a"*r*: "r" in "R"};

* a "two-sided principal ideal" is a subset of the form "R"*a*"R" := {"r"_{1}"a"*s*_{1}+ ... + "r"_{"n"}"a"*s*_{"n"}: "r"_{1},"s"_{1},...,"r"_{"n"},"s"_{"n"}in "R"}If "R" is acommutative ring , then the above three notions are all the same.In that case, it is common to write the ideal generated by "a" as ("a").Not all ideals are principal.For example, consider the commutative ring

**C**["x","y"] of allpolynomial s in twovariable s "x" and "y", with complex coefficients.The ideal ("x","y") generated by "x" and "y", which consists of all the polynomials in**C**["x","y"] that have zero for theconstant term , is not principal.To see this, suppose that "p" were a generator for ("x","y"); then "x" and "y" would both be divisible by "p", which is impossible unless "p" is a nonzero constant.But zero is the only constant in ("x","y"), so we have acontradiction .A ring in which every ideal is principal is called "principal", or a

principal ideal ring .Aprincipal ideal domain (PID) is anintegral domain that is principal.Any PID must be aunique factorization domain ; the normal proof of unique factorization in theinteger s (the so-calledfundamental theorem of arithmetic ) holds in any PID.Also, any

Euclidean domain is a PID; the algorithm used to calculategreatest common divisor s may be used to find a generator of any ideal.More generally, any two principal ideals in a commutative ring have a greatest common divisor in the sense of ideal multiplication.In principal ideal domains, this allows us to calculate greatest common divisors of elements of the ring, up to multiplication by a unit; we define gcd("a","b") to be any generator of the ideal ("a","b").For a

Dedekind domain "R", we may also ask, given a non-principal ideal "I" of "R", whether there is some extension "S" of "R" such that the ideal of "S" generated by "I" is principal (said more loosely, "I" "becomes principal" in "S"). This question arose in connection with the study of rings ofalgebraic integer s (which are examples of Dedekind domains) innumber theory , and led to the development ofclass field theory byTeiji Takagi ,Emil Artin ,David Hilbert , and many others.The principal ideal theorem of class field theory states that every integer ring "R" (i.e. the

ring of integers of somenumber field ) is contained in a larger integer ring "S" which has the property that "every" ideal of "R" becomes a principal ideal of "S".In this theorem we may take "S" to be the ring of integers of theHilbert class field of "R"; that is, the maximal unramified abelian extension (that is, Galois extension whose Galois group is abelian) of the fraction field of "R", and this is uniquely determined by "R".Krull's principal ideal theorem states that if "R" is a Noetherian ring and "I" is a principal, proper ideal of "R", then "I" has height at most one.**References*** cite book

title = Contemporary Abstract Algebra

author = Joseph A. Gallian

publisher = Houghton Mifflin

year = 2004

pages = 262

isbn = 9780618514717

accessdate = 2008-03-26

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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] * * * … 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**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