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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:
If R is a commutative 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 all polynomials in two variables 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 the constant 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 a contradiction.
A ring in which every ideal is principal is called principal, or a principal ideal ring. A principal ideal domain (PID) is an integral domain that is principal. Any PID must be a unique factorization domain; the normal proof of unique factorization in the integerThe integers consist of the positive natural numbers (1, 2, 3, …) the negative natural numbers (−1, −2, −3,. and the number zero. The set of all integers is usually denoted in mathematics by Z (or Z in blackboard bold, ), which sts (the so-called fundamental theorem of arithmeticIn mathematics, and in particular number theory, the fundamental theorem of arithmetic or unique factorization theorem is the statement that every positive integer can be written as a product of prime numbers in only one way. For instance, we can write :6) holds in any PID.
Also, any Euclidean domainIn abstract algebra, a Euclidean domain (also called a Euclidean ring is a type of ring in which the Euclidean algorithm can be used. More precisely, a Euclidean domain is an integral domain D for which can be defined a function v mapping nonzero elements is a PID; the algorithm used to calculate greatest common divisorNumber theory In mathematics, the greatest common divisor (gcd), sometimes known as the greatest common factor (GCF) or highest common factor (hcf) of two integers which are not both zero is the largest integer that divides both numbers. The greatest comms may be used to find a generator of any ideal. More genarally, 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 unitRing theory In mathematics, a unit in a ring R is an element u such that there is v in R with uv vu 1. That is, u is an invertible element of the multiplicative monoid of R''. The units of R form a group U ''R under multiplication, the group of units of R; we define gcd(a,b) to be any generator of the ideal (a,b).
For a Dedekind domainCommutative algebra Algebraic number theory In abstract algebra, a Dedekind domain is a Noetherian integral domain which is integrally closed in its fraction field and which has Krull dimension 1. In other words, a Dedekind domain is a commutative ring wh 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 of algebraic integerAlgebraic number theory In mathematics, an algebraic integer is a complex number α that is a root of an equation P ''x 0 where P ''x is a monic polynomial with integer coefficients. The algebraic integers are all therefore algebraic numbers, but nots (which are examples of Dedekind domains) in number theoryTraditionally, number theory is that branch of pure mathematics concerned with the properties of integers and contains many open problems that are easily understood even by non-mathematicians. More generally, the field has come to be concerned with a wide, and led to the development of class field theory by Teiji Takagi, Emil Artin, David Hilbert, and many others. It turns out that every integer ring R (i.e. the ring of algebraic integers of some number field) is contained in a larger integer ring S which has the property that every ideal of R becomes a principal ideal of S.
The fraction field of S is then called the Hilbert class field of R; it is the maximal unramified abelian extension (that is, Galois extension whose Galois group is abelian) of the fraction field of R, and it is uniquely determined by R.
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