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. In mathematics, ring theory is the study of rings, algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers. ... Abstract algebra is the field of mathematics concerned with the study of algebraic structures such as groups, rings and fields. ... In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have similar (but not identical) properties to those familiar from the integers. ...

More specifically:

• a left principal ideal of R is a subset of R of the form Ra := {ra : r in R};
• a right principal ideal is a subset of the form aR := {ar : r in R};
• a two-sided principal ideal is a subset of the form RaR := {r₁ar'₁ + ... + r_nar'_n : r₁,r'₁,...,r_n,r'_n in R}

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). A is a subset of B If X and Y are sets and every element of X is also an element of Y, then we say or write: X is a subset of (or is included in) Y; X ⊆ Y; Y is a superset of (or includes) X; Y ⊇ X... In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ...

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. In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. ... In computer science and mathematics, a variable is a symbol denoting a quantity or symbolic representation. ... In mathematics, the complex numbers are an extension of the real numbers by the inclusion of the imaginary unit i, satisfying . ... 0 (zero) or nought is both a number and a numeral. ... In mathematics and the mathematical sciences, a constant is a fixed, but possibly unspecified, value. ... Broadly speaking, a contradiction is when two or more statements, ideas, or actions are seen as incompatible. ...

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 integers (the so-called fundamental theorem of arithmetic) holds in any PID. In abstract algebra, a principal ideal domain (PID) is an integral domain in which every ideal is principal (that is, generated by a single element). ... In abstract algebra, an integral domain is a commutative ring with 0 ≠ 1 in which the product of any two non-zero elements is always non-zero. ... In mathematics, a unique factorization domain (UFD) is, roughly speaking, a commutative ring in which every element can be uniquely written as a product of prime elements, analogous to the fundamental theorem of arithmetic for the integers. ... The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ... In mathematics, and in particular number theory, the fundamental theorem of arithmetic or unique factorization theorem is the statement that every positive integer greater than 1 can be written as a product of prime numbers in only one way. ...

Also, any Euclidean domain is a PID; the algorithm used to calculate greatest common divisors 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). In abstract algebra, a Euclidean domain (also called a Euclidean ring) is a type of ring in which the Euclidean algorithm can be used. ... 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. ... In mathematics, a unit in a ring R is an element u such that there is v in R with uv = vu = 1R. 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...

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 of algebraic integers (which are examples of Dedekind domains) in number theory, 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. 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 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 (that is, the coefficient of the largest power of x in P(x) is one) with integer coefficients. ... Traditionally, number theory is that branch of pure mathematics concerned with the properties of integers. ... Class field theory is a branch of algebraic number theory, including most of the major results that were proved in the period about 1900-1950. ... Teiji Takagi (高木 貞治 Takagi Teiji, April 21, 1875 - February 28, 1960) was a Japanese mathematician, best known for proving the Takagi existence theorem in class field theory. ... Emil Artin (March 3, 1898-December 20, 1962) was a mathematician born in Vienna, Austria who spent his career in Germany (mainly in Hamburg) until the Nazi threat when he emigrated to the USA in 1937 where he was at Indiana University 1938-1946, and Princeton University 1946-1958. ... David Hilbert David Hilbert (January 23, 1862 – February 14, 1943) was a German mathematician born in Wehlau, near Königsberg, Prussia (now Znamensk, near Kaliningrad, Russia) who is recognized as one of the most influential mathematicians of the 19th and early 20th centuries. ... In mathematics, an algebraic number field (or simply number field) is a finite field extension of the rational numbers Q. That is, it is a field which contains Q and has finite dimension when considered as a vector space over Q. The study of algebraic number fields, and these days...

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. In mathematics, every integral domain can be embedded in a field; the smallest field which can be used is the quotient field or the field of fractions of the integral domain. ... In mathematics, ramification is a geometric term used for branching out, in the way that the square root function, for complex numbers, can be seen to have two branches differing in sign. ... In mathematics, Galois theory is a branch of abstract algebra. ... In mathematics, an abelian group, also called a commutative group, is a group (G, *) such that a * b = b * a for all a and b in G. Abelian groups are named after Niels Henrik Abel. ...