prime ideal
Let be a ring. A two-sided proper ideal of a ring is called a prime ideal
if the following equivalent
conditions are met:
- 1.
If and are left ideals
and the product of ideals satisfies , then or .
- 2.
If and are right ideals with , then or .
- 3.
If and are two-sided ideals with , then or .
- 4.
If and are elements of with , then or .
is a prime ring if and only if is a prime ideal. When is commutative
with identity
, a proper ideal of is prime if and only if for any , if then either or . One also has in this case that is prime if and only if the quotient ring
is an integral domain.