Editing Prime element

{{short description|Analogue of a prime number in a commutative ring}}
In [[mathematics]], specifically in [[abstract algebra]], a '''prime element''' of a [[commutative ring]] is an object satisfying certain properties similar to the [[prime number]]s in the [[integer]]s and to [[irreducible polynomial]]s. Care should be taken to distinguish prime elements from [[irreducible element]]s, a concept that is the same in [[unique factorization domain|UFD]]s but not the same in general.

==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.

==Connection with prime ideals==
{{Main|Prime ideal}}
An ideal {{math|''I''}} in the ring {{math|''R''}} (with unity) is [[prime ideal|prime]] if the factor ring {{math|''R''/''I''}} is an [[integral domain]]. Equivalently, {{math|''I''}} is prime if whenever <math>ab \in I</math> then either <math>a \in I</math> or <math>b \in I</math>. 

In an integral domain, a nonzero [[principal ideal]] is [[prime ideal|prime]] if and only if it is generated by a prime element.

==Irreducible elements==
{{Main|Irreducible element}}
Prime elements should not be confused with [[irreducible element]]s.  In an [[integral domain]], every prime is irreducible<ref>{{harvnb|Hungerford|1980|loc=Theorem III.3.4(iii)}}</ref> but the converse is not true in general.  However, in unique factorization domains,<ref>{{harvnb|Hungerford|1980|loc=Remark after Definition III.3.5}}</ref> or more generally in [[GCD domain]]s, primes and irreducibles are the same.

==Examples==
The following are examples of prime elements in rings:
* The integers {{math|±2}}, {{math|±3}}, {{math|±5}}, {{math|±7}}, {{math|±11}}, ... in the [[ring of integers]] {{math|'''Z'''}}
* the complex numbers {{math|(1 + ''i'')}}, {{math|19}}, and {{math|(2 + 3''i'')}} in the ring of [[Gaussian integers]] {{math|'''Z'''[''i'']}}
* the polynomials {{math|''x''<sup>2</sup> − 2}} and {{math|''x''<sup>2</sup> + 1}} in {{math|'''Z'''[''x'']}}, the [[ring of polynomials]] over {{math|'''Z'''}}.
* 2 in the [[quotient ring]] {{math|'''Z'''/6'''Z'''}}
* {{math|''x''<sup>2</sup> + (''x''<sup>2</sup> + ''x'')}} is prime but not irreducible in the ring {{math|'''Q'''[''x'']/(''x''<sup>2</sup> + ''x'')}}
* In the ring {{math|'''Z'''<sup>2</sup>}} of pairs of integers, {{math|(1, 0)}} is prime but not irreducible (one has {{math|1=(1, 0)<sup>2</sup> = (1, 0)}}).
* In the [[ring of algebraic integers]] <math>\mathbf Z[\sqrt{-5}],</math> the element {{math|3}} is irreducible but not prime (as 3 divides <math>9=(2+\sqrt{-5})(2-\sqrt{-5})</math> and 3 does not divide any factor on the right).

==References==
;Notes
{{reflist}}

;Sources
*Section III.3 of {{Citation
| authorlink=Thomas W. Hungerford
| last=Hungerford
| first=Thomas W.
| title=Algebra
| edition=Reprint of 1974
| year=1980
| publisher=[[Springer-Verlag]]
| location=New York
| series=[[Graduate Texts in Mathematics]]
| volume=73
| isbn=978-0-387-90518-1
| mr=0600654
}}
*{{citation   |author=[[Nathan Jacobson|Jacobson, Nathan]]   |title=Basic algebra. II   |edition=2   |publisher=W. H. Freeman and Company   |place=New York   |year=1989   |pages=xviii+686   |isbn=0-7167-1933-9   |mr=1009787}}
*{{citation   |author=[[Irving Kaplansky|Kaplansky, Irving]]   |title=Commutative rings   |publisher=Allyn and Bacon Inc.   |place=Boston, Mass.   |year=1970   |pages=x+180   |mr=0254021 }}

[[Category:Ring theory]]