prime ideal (original) (raw)

Let R be a ring. A two-sided proper ideal 𝔭 of a ring R is called a prime ideal if the following equivalent conditions are met:

1. 1. If I and J are left ideals and the product of ideals I⁢J satisfies I⁢J⊂𝔭, then I⊂𝔭 or J⊂𝔭.
2. 1. If I and J are right ideals with I⁢J⊂𝔭, then I⊂𝔭 or J⊂𝔭.
3. 1. If I and J are two-sided ideals with I⁢J⊂𝔭, then I⊂𝔭 or J⊂𝔭.
4. 1. If x and y are elements of R with x⁢R⁢y⊂𝔭, then x∈𝔭 or y∈𝔭.

R/𝔭 is a prime ring if and only if 𝔭 is a prime ideal. When R is commutative with identity, a proper ideal 𝔭 of R is prime if and only if for any a,b∈R, if a⋅b∈𝔭 then either a∈𝔭 or b∈𝔭. One also has in this case that 𝔭⊂R is prime if and only if the quotient ring R/𝔭 is an integral domain.