Principal ideal domain
In mathematics, 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. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are principal, although some authors (e.g., Bourbaki) refer to PIDs as principal rings. The distinction is that a principal ideal ring may have zero divisors whereas a principal ideal domain cannot. Principal ideal domains are thus mathematical objects that behave somewhat like the integers, with respect to divisibility: any element of a PID has a unique decomposition  into prime elements (so an analogue of the fundamental theorem of arithmetic holds); any two elements of a PID have a greatest common divisor (although it may not be possible to find it using the Euclidean algorithm). If and are elements of a PID without common divisors, then every element of the PID can be written in the form . Principal ideal domains are noetherian, they are integrally closed, they are unique factorization domains and Dedekind domains. All Euclidean domains and all fields are principal ideal domains. Principal ideal domains appear in the following chain of class inclusions: Examples. Examples include : Non-examples. Examples of integral domains that are not PIDs: Modules. The key result is the structure theorem: If "R" is a principal ideal domain, and "M" is a finitely generated "R"-module, then formula_20 is a direct sum of cyclic modules, i.e., modules with one generator. The cyclic modules are isomorphic to formula_21 for some formula_22 (notice that formula_23 may be equal to formula_24, in which case formula_21 is formula_26). If "M" is a free module over a principal ideal domain "R", then every submodule of "M" is again free. This does not hold for modules over arbitrary rings, as  the example formula_27 of modules over formula_28 shows. Properties. In a principal ideal domain, any two elements have a greatest common divisor, which may be obtained as a generator of the ideal . All Euclidean domains are principal ideal domains, but the converse is not true. An example of a principal ideal domain that is not a Euclidean domain is the ring formula_29 In this domain no and exist, with , so that formula_30, despite formula_31 and formula_32 having a greatest common divisor of . Every principal ideal domain is a unique factorization domain (UFD). The converse does not hold since for any  UFD , the ring of polynomials in 2 variables is a UFD but is not a PID. (To prove this look at the ideal generated by formula_33 It is not the whole ring since it contains no polynomials of degree 0, but it cannot be generated by any one single element.) The previous three statements give the definition of a Dedekind domain, and hence every principal ideal domain is a Dedekind domain. Let "A" be an integral domain. Then the following are equivalent. Any Euclidean norm is a Dedekind-Hasse norm; thus, (5) shows that a Euclidean domain is a  PID. (4) compares to: An integral domain is a Bézout domain if and only if any two elements in it have a gcd "that is a linear combination of the two." A Bézout domain is thus a GCD domain, and (4) gives yet another proof that a PID is a UFD.