Editing Prime ideal (section)

=== Non-examples ===
* Consider the [[function composition|composition]] of the following two [[quotient ring|quotients]]
::<math>\Complex[x,y] \to \frac{\Complex[x,y]}{(x^2 + y^2 - 1)} \to \frac{\Complex[x,y]}{(x^2 + y^2 - 1, x)}</math>
:Although the first two rings are integral domains (in fact the first is a UFD) the last is not an integral domain since it is [[ring homomorphism|isomorphic]] to
::<math>\frac{\Complex[x,y]}{(x^2 + y^2 - 1, x)} \cong \frac{\Complex[y]}{(y^2 - 1)} \cong \Complex\times\Complex</math>
:since <math>(y^2 - 1)</math> factors into <math>(y - 1)(y + 1)</math>, which implies the existence of [[Zero_divisor|zero divisors]] in the quotient ring, preventing it from being isomorphic to <math>\Complex</math> and instead to non-integral domain <math>\Complex\times\Complex</math> (by the [[Chinese_remainder_theorem#Statement|Chinese remainder theorem]]).
:This shows that the ideal <math>(x^2 + y^2 - 1, x) \subset \Complex[x,y]</math> is not prime. (See the first property listed below.)

* Another non-example is the ideal <math>(2,x^2 + 5) \subset \Z[x]</math> since we have
::<math>x^2+5 -2\cdot 3=(x-1)(x+1)\in (2,x^2+5)</math> 
:but neither <math>x-1</math> nor <math>x+1</math> are elements of the ideal.