- Prime ideal
In

mathematics , a**prime ideal**is asubset of a ring which shares many important properties of aprime number in thering of integers . This article only covers ideals of ring theory. Prime ideals inorder theory are treated in the article on ideals in order theory.**Formal definition*** An ideal "P" of a ring "R" is prime if and only if it is a proper ideal (ie "P" ≠ "R") and for any two ideals "A" and "B" in "R" such that "AB" ⊆ "P", either "A" ⊆ "P" or "B" ⊆ "P".

This is close to the historical point of view of ideals as ideal numbers, as for the ring

**Z**"A" is contained in "P" is another way of saying "P" divides "A", and the unit ideal "R" represents unity.**Prime ideals for commutative rings**Prime ideals have a simpler description for

commutative ring s: if "R" is a commutative ring, then an ideal "P" of "R" is "prime" if it has the following two properties:

* whenever "a", "b" are two elements of "R" such that their product "ab" lies in "P", then "a" is in "P" or "b" is in "P".

* "P" is not equal to the whole ring "R"This generalizes the following property of prime numbers: if "p" is a prime number and if "p" divides a product "ab" of twointeger s, then "p" divides "a" or "p" divides "b". We can therefore say:A positive integer "n" is a prime number if and only if the ideal "n**"Z**is a prime ideal in**Z**.**Examples*** If "R" denotes the ring

**C**["X", "Y"] of polynomials in two variables with complex coefficients, then the ideal generated by the polynomial "Y"^{2}− "X"^{3}− "X" − 1 is a prime ideal (seeelliptic curve ).

* In the ring**Z**["X"] of all polynomials with integer coefficients, the ideal generated by 2 and "X" is a prime ideal. It consists of all those polynomials whose constant coefficient is even.

* In any ring "R", a**maximal ideal**is an ideal "M" that is maximal in the set of all proper ideals of "R", i.e. "M" is contained in exactly 2 ideals of "R", namely "M" itself and the entire ring "R". Every maximal ideal is in fact prime; in aprincipal ideal domain every nonzero prime ideal is maximal, but this is not true in general.

* If "M" is a smoothmanifold , "R" is the ring of smooth functions on "M", and "x" is a point in "M", then the set of all smooth functions "f" with "f"("x") = 0 forms a prime ideal (even a maximal ideal) in "R".**Properties*** An ideal "I" in the commutative ring "R" is prime if and only if the factor ring "R/I" is an

integral domain .

* An ideal "I" of a ring "R" is prime if and only if "R" "I" is closed under multiplication.

* Every nonzero commutative ring contains at least one prime ideal (in fact it contains at least one maximal ideal) which is a direct consequence ofKrull's theorem

* A commutative ring is anintegral domain if and only if {0} is a prime ideal.

* A commutative ring is a field if and only if {0} is its only prime ideal, or equivalently, if and only if {0} is a maximal ideal.

* Thepreimage of a prime ideal under a ring homomorphism is a prime ideal

* The sum of two prime ideals is not necessairly prime. For an example, consider the ring $mathbb\{C\}\; [x,y]$ with prime ideals P =(x^2+y^2-1) and Q = (x) (the ideals generated by x^2 + y^2 - 1 and x respectively). Their sum P + Q = (x^2+y^2-1,x)=(y^2-1,x) however is not prime. To see this note the quotient ring has zero divisors implying that the quotient is not an integral domain and thus P + Q cannot be prime.**Uses**One use of prime ideals occurs in

algebraic geometry , where varieties are defined as the zero sets of ideals in polynomial rings. It turns out that the irreducible varieties correspond to prime ideals. In the modern abstract approach, one starts with an arbitrary commutative ring and turns the set of its prime ideals, also called its spectrum, into atopological space and can thus define generalizations of varieties called schemes, which find applications not only ingeometry , but also innumber theory .The introduction of prime ideals in

algebraic number theory was a major step forward: it was realized that the important property of unique factorisation expressed in thefundamental theorem of arithmetic does not hold in every ring ofalgebraic integer s, but a substitute was found whenDedekind replaced elements by ideals and prime elements by prime ideals; seeDedekind domain .**Prime ideals for noncommutative rings**If "R" is a noncommutative ring, then an ideal "P" of "R" is "prime" if it has the following two properties:

* whenever "a", "b" are two elements of "R" such that for all elements "r" of "R", their product "arb" lies in "P", then "a" is in "P" or "b" is in "P".

* "P" is not equal to the whole ring "R".For commutative rings this definition is equivalent to the one given in the previous section. For noncommutative rings, the two definitions are different. An ideal such that "ab" in "P" implies that "a" or "b" is in "P" is called a

**completely prime ideal**. Completely prime ideals are prime ideals, but the converse is not true. For example, the zero ideal in the ring of "n" × "n" matrices is a prime ideal, but it is not completely prime.**Examples*** Any

maximal ideal is prime.* Any

primitive ideal is prime.* The zero ideal of any

prime ring is prime.

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