Editing Integral domain (section)

== Divisibility, prime elements, and irreducible elements ==
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{{see also|Divisibility (ring theory)}}
In this section, ''R'' is an integral domain.

Given elements ''a'' and ''b'' of ''R'', one says that ''a'' ''divides'' ''b'', or that ''a'' is a ''[[Divisibility (ring theory)|divisor]]'' of ''b'', or that ''b'' is a ''multiple'' of ''a'', if there exists an element ''x'' in ''R'' such that {{nowrap|1=''ax'' = ''b''}}.

The ''[[unit (ring theory)|unit]]s'' of ''R'' are the elements that divide 1; these are precisely the invertible elements in ''R''. Units divide all other elements.

If ''a'' divides ''b'' and ''b'' divides ''a'', then ''a'' and ''b'' are '''associated elements''' or '''associates'''.{{sfn|Durbin|1993|loc=p. 224, "Elements ''a'' and ''b'' of [an integral domain] are called ''associates'' if ''a'' {{!}} ''b'' and ''b'' {{!}} ''a''."|ps=none}}  Equivalently, ''a'' and ''b'' are associates if {{nowrap|1=''a'' = ''ub''}} for some [[unit (ring theory)|unit]] ''u''.

An ''[[irreducible element]]'' is a nonzero non-unit that cannot be written as a product of two non-units.

A nonzero non-unit ''p'' is a ''[[prime element]]'' if, whenever ''p'' divides a product ''ab'', then ''p'' divides ''a'' or ''p'' divides ''b''. Equivalently, an element ''p'' is prime if and only if the [[principal ideal]] (''p'') is a nonzero [[prime ideal]]. 

Both notions of irreducible elements and prime elements generalize the ordinary definition of [[prime number]]s in the ring <math>\Z,</math> if one considers as prime the negative primes.

Every prime element is irreducible. The converse is not true in general: for example, in the [[quadratic integer]] ring <math>\Z\left[\sqrt{-5}\right]</math> the element 3 is irreducible (if it factored nontrivially, the factors would each have to have norm 3, but there are no norm 3 elements since <math>a^2+5b^2=3</math> has no integer solutions), but not prime (since 3 divides <math>\left(2 + \sqrt{-5}\right)\left(2 - \sqrt{-5}\right)</math> without dividing either factor). In a unique factorization domain (or more generally, a [[GCD domain]]), an irreducible element is a prime element.

While [[Fundamental theorem of arithmetic|unique factorization]] does not hold in <math>\Z\left[\sqrt{-5}\right]</math>, there is unique factorization of [[Ideal (ring theory)|ideals]].  See [[Lasker–Noether theorem]].