prime integer can no longer be factored into elements, instead they are prime ideals. In fact many for the p-th root of unity are not principal ideal domains . In fact, the class number of a ring of algebraic integers gives a notion of "how far away" it is from being a principal ideal domain. Modules The key result is the structure theorem: If R is a principal ideal domain, and M is a finitely generated R-module, then is a direct sum of cyclic modules, i.e., modules with one generator. The cyclic modules are isomorphic to for some (notice that may be equal to , in which case is ). 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 of modules over 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 In this domain no and exist, with , so that , despite and 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 It is not the whole ring since it contains no polynomials of degree 0, but it cannot be generated by any one single element.) Every principal ideal domain is Noetherian. In all unital rings, maximal ideals are prime. In principal ideal domains a near converse holds: every nonzero prime ideal is maximal. All principal ideal domains are integrally closed. 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. A is a PID. Every prime ideal of A is principal. A is a Dedekind domain that is a UFD. Every finitely generated ideal of A is principal (i.e., A is a Bézout domain) and A satisfies the ascending chain condition on principal ideals. A