Editing Prime ideal (section)

===Properties===
* An ideal {{math|''I''}} in the ring {{mvar|R}} (with [[unital ring|unity]]) is prime if and only if the [[factor ring]] {{math|''R''/''I''}} is an [[integral domain]]. In particular, a commutative ring (with unity) is an integral domain if and only if {{math|(0)}} is a prime ideal.  (The [[zero ring]] has no prime ideals, because the ideal (0) is the whole ring.)
* An ideal {{math|''I''}} is prime if and only if its set-theoretic [[complement (set theory)|complement]] is [[multiplicatively closed set|multiplicatively closed]].<ref>{{cite book | last=Reid | first=Miles | author-link=Miles Reid | title=Undergraduate Commutative Algebra | publisher=[[Cambridge University Press]] | year=1996 | isbn=0-521-45889-7}}</ref>
* Every nonzero ring contains at least one prime ideal (in fact it contains at least one maximal ideal), which is a direct consequence of [[Krull's theorem]].
* More generally, if {{mvar|S}} is any multiplicatively closed set in {{mvar|R}}, then a lemma essentially due to Krull shows that there exists an ideal of {{mvar|R}} maximal with respect to being [[disjoint sets|disjoint]] from {{mvar|S}}, and moreover the ideal must be prime. This can be further generalized to noncommutative rings (see below).<ref name="Lam">Lam ''First Course in Noncommutative Rings'', p. 156</ref> In the case {{math|''S'' {{=}} {1},}} we have [[Krull's theorem]], and this recovers the maximal ideals of {{mvar|R}}. Another prototypical m-system is the set, {{math|{''x'', ''x''<sup>2</sup>, ''x''<sup>3</sup>, ''x''<sup>4</sup>, ...},}} of all positive powers of a non-[[nilpotent]] element.
* The [[preimage]] of a prime ideal under a [[ring homomorphism]] is a prime ideal.  The analogous fact is not always true for [[maximal ideal]]s, which is one reason algebraic geometers define the [[spectrum of a ring]] to be its set of prime rather than maximal ideals; one wants a homomorphism of rings to give a map between their spectra.
* The set of all prime ideals (called the [[spectrum of a ring]]) contains minimal elements (called [[minimal prime ideal]]s). Geometrically, these correspond to irreducible components of the spectrum.
* The sum of two prime ideals is not necessarily prime.  For an example, consider the ring <math>\Complex[x,y]</math> with prime ideals {{math|''P'' {{=}} (''x''<sup>2</sup> + ''y''<sup>2</sup> − 1)}} and {{math|''Q'' {{=}} (''x'')}} (the ideals generated by {{math|''x''<sup>2</sup> + ''y''<sup>2</sup> − 1}} and {{math|''x''}} respectively). Their sum {{math|''P'' + ''Q'' {{=}} (''x''<sup>2</sup> + ''y''<sup>2</sup> − 1, ''x'') {{=}} (''y''<sup>2</sup> − 1, ''x'')}} however is not prime: {{math|''y''<sup>2</sup> − 1 {{=}} (''y'' − 1)(''y'' + 1) ∈ ''P'' + ''Q''}} but its two factors are not. Alternatively, the quotient ring has [[zero divisor]]s so it is not an integral domain and thus {{math|''P'' + ''Q''}} cannot be prime. 
* Not every ideal which cannot be factored into two ideals is a prime ideal; e.g. <math> (x,y^2)\subset \mathbb{R}[x,y]</math> cannot be factored but is not prime.
* In a commutative ring {{mvar|R}} with at least two elements, if every proper ideal is prime, then the ring is a field. (If the ideal {{math|(0)}} is prime, then the ring {{mvar|R}} is an integral domain. If {{mvar|q}} is any non-zero element of {{mvar|R}} and the ideal {{math|(''q''<sup>2</sup>)}} is prime, then it contains {{mvar|q}} and then {{mvar|q}} is [[unit (ring theory)|invertible]].)
* A nonzero principal ideal is prime if and only if it is generated by a [[prime element]]. In a UFD, every nonzero prime ideal contains a prime element.