Editing Prime ideal (section)

==Prime ideals for noncommutative rings==
The notion of a prime ideal can be generalized to noncommutative rings by using the commutative definition "ideal-wise". [[Wolfgang Krull]] advanced this idea in 1928.<ref>Krull, Wolfgang, ''Primidealketten in allgemeinen Ringbereichen'', Sitzungsberichte Heidelberg. Akad. Wissenschaft (1928), 7. Abhandl.,3-14.</ref> The following content can be found in texts such as Goodearl's<ref>Goodearl, ''An Introduction to Noncommutative Noetherian Rings''</ref> and Lam's.<ref>Lam, ''First Course in Noncommutative Rings''</ref> If {{mvar|R}} is a (possibly noncommutative) ring and {{mvar|P}} is a proper ideal of {{mvar|R}}, we say that {{mvar|P}} is '''prime''' if for any two ideals {{mvar|A}} and {{mvar|B}} of {{mvar|R}}:

* If the product of ideals {{math|''AB''}} is contained in {{mvar|P}}, then at least one of {{mvar|A}} and {{mvar|B}} is contained in {{mvar|P}}.

It can be shown that this definition is equivalent to the commutative one in commutative rings. It is readily verified that if an ideal of a noncommutative ring {{mvar|R}} satisfies the commutative definition of prime, then it also satisfies the noncommutative version. An ideal {{mvar|P}} satisfying the commutative definition of prime is sometimes called a '''completely prime ideal''' to distinguish it from other merely prime ideals in the ring.  Completely prime ideals are prime ideals, but the [[converse (logic)|converse]] is not true.  For example, the zero ideal in the ring of {{math|''n''&thinsp;×&thinsp;''n''}} [[matrix (mathematics)|matrices]] over a field is a prime ideal, but it is not completely prime.

This is close to the historical point of view of ideals as [[ideal number]]s, as for the ring <math>\Z</math> "{{mvar|A}} is contained in {{mvar|P}}" is another way of saying "{{mvar|P}} divides {{mvar|A}}", and the unit ideal {{mvar|R}} represents unity.

Equivalent formulations of the ideal {{math|''P'' ≠ ''R''}} being prime include the following properties:
* For all {{mvar|a}} and {{mvar|b}} in {{mvar|R}}, {{math|(''a'')(''b'') ⊆ ''P''}} implies {{math|''a'' ∈ ''P''}} or {{math|''b'' ∈ ''P''}}.
* For any two ''right'' ideals of {{mvar|R}}, {{math|''AB'' ⊆ ''P''}} implies {{math|''A'' ⊆ ''P''}} or {{math|''B'' ⊆ ''P''}}.
* For any two ''left'' ideals of {{mvar|R}}, {{math|''AB'' ⊆ ''P''}} implies {{math|''A'' ⊆ ''P''}} or {{math|''B'' ⊆ ''P''}}.
* For any elements {{mvar|a}} and {{mvar|b}} of {{mvar|R}}, if {{math|''aRb'' ⊆ ''P''}}, then {{math|''a'' ∈ ''P''}} or {{math|''b'' ∈ ''P''}}.

Prime ideals in commutative rings are characterized by having multiplicatively closed [[complement (set theory)|complements]] in {{mvar|R}}, and with slight modification, a similar characterization can be formulated for prime ideals in noncommutative rings. A [[empty set|nonempty]] subset {{math|''S'' ⊆ ''R''}} is called an '''m-system''' if for any {{mvar|a}} and {{mvar|b}} in {{mvar|S}}, there exists {{mvar|r}} in {{mvar|R}} such that {{math|''arb''}} is in {{mvar|S}}.<ref>Obviously, multiplicatively closed sets are m-systems.</ref> The following item can then be added to the list of equivalent conditions above:

* The complement {{math|''R''∖''P''}} is an m-system.

===Examples===
* Any [[primitive ideal]] is prime.
* As with commutative rings, maximal ideals are prime, and also prime ideals contain minimal prime ideals.
* A ring is a [[prime ring]] if and only if the zero ideal is a prime ideal, and moreover a ring is a [[integral domain|domain]] if and only if the zero ideal is a completely prime ideal.
* Another fact from commutative theory echoed in noncommutative theory is that if {{mvar|A}} is a nonzero {{mvar|R}}-[[module (mathematics)|module]], and {{mvar|P}} is a maximal element in the [[poset]] of [[Annihilator (ring theory)|annihilator]] ideals of submodules of {{mvar|A}}, then {{mvar|P}} is prime.