Editing Commutative ring (section)

==== Principal ideal domains ====
If <math> F </math> consists of a single element <math> r </math>, the ideal generated by <math> F </math> consists of the multiples of <math> r </math>, i.e., the elements of the form <math> rs </math> for arbitrary elements <math> s </math>. Such an ideal is called a [[principal ideal]]. If every ideal is a principal ideal, <math> R </math> is called a [[principal ideal ring]]; two important cases are <math> \mathbb{Z} </math> and <math> k \left[X\right] </math>, the polynomial ring over a field <math> k </math>. These two are in addition domains, so they are called [[principal ideal domain]]s.

Unlike for general rings, for a principal ideal domain, the properties of individual elements are strongly tied to the properties of the ring as a whole. For example, any principal ideal domain <math> R </math> is a [[unique factorization domain]] (UFD) which means that any element is a product of irreducible elements, in a (up to reordering of factors) unique way. Here, an element <math> a </math> in a domain is called [[irreducible element|irreducible]] if the only way of expressing it as a product
<math display="block"> a=bc ,</math>
is by either <math> b </math> or <math> c </math> being a unit. An example, important in [[Field (mathematics)|field theory]], are [[irreducible polynomial]]s, i.e., irreducible elements in <math> k \left[X\right] </math>, for a field <math> k </math>. The fact that <math> \mathbb{Z} </math> is a UFD can be stated more elementarily by saying that any natural number can be uniquely decomposed as product of powers of prime numbers. It is also known as the [[fundamental theorem of arithmetic]].

An element <math> a </math> is a [[prime element]] if whenever <math> a </math> divides a product <math> bc </math>, <math> a </math> divides <math> b </math> or <math> c </math>. In a domain, being prime implies being irreducible. The converse is true in a unique factorization domain, but false in general.