Editing Prime element (section)

==Definition==
An element {{mvar|p}} of a commutative ring {{mvar|R}} is said to be '''prime''' if it is not the [[zero element]] or a [[unit (ring theory)|unit]] and whenever {{mvar|p}} [[Divisibility (ring theory)|divides]] {{mvar|ab}} for some {{mvar|a}} and {{mvar|b}} in {{mvar|R}}, then {{mvar|p}} divides {{mvar|a}} or {{mvar|p}} divides {{mvar|b}}. With this definition, [[Euclid's lemma]] is the assertion that [[prime number]]s are prime elements in the [[ring of integers]].  Equivalently, an element {{mvar|p}} is prime if, and only if, the [[principal ideal]] {{math|(''p'')}} generated by {{mvar|p}} is a nonzero [[prime ideal]].<ref>{{harvnb|Hungerford|1980|loc=Theorem III.3.4(i)}}, as indicated in the remark below the theorem and the proof, the result holds in full generality.</ref> (Note that in an [[integral domain]], the ideal {{math|(0)}} is a [[prime ideal]], but {{math|0}} is an exception in the definition of 'prime element'.)

Interest in prime elements comes from the [[fundamental theorem of arithmetic]], which asserts that each nonzero integer can be written in essentially only one way as 1 or −1 multiplied by a product of positive prime numbers. This led to the study of [[unique factorization domain]]s, which generalize what was just illustrated in the integers.

Being prime is relative to which ring an element is considered to be in; for example, 2 is a prime element in {{math|'''Z'''}} but it is not in {{math|'''Z'''[''i'']}}, the ring of [[Gaussian integers]], since {{math|1=2 = (1 + ''i'')(1 − ''i'')}} and 2 does not divide any factor on the right.