General Ring Theory/Prime ideals

Definition (prime ideal):

Let be a ring. A prime ideal of is an ideal such that whenever are ideals of so that , then either or .

Definition (prime ring):

A ring is called a prime ring iff its zero ideal is a prime ideal.

Proposition (characterisation of prime ideals):

Let be a ring, and let be an ideal. The following are equivalent:

  1. is a prime ideal of
  2. is a prime ring
  3. Whenever are left ideals in , then
  4. Whenever are right ideals in , then
  5. Whenever are such that , then either or .

Proof: We'll prove , since follows by symmetry. Suppose first that is a prime ideal. Let so that , the zero ideal of . Then if is the projection, consider , . Then (since is a ring homomorphism), so that without loss of generality , and hence . Suppose now that is a prime ring. Let then such that . We use the bar notation ( being the projection) for . Then we get that is zero for all . Then define the ideals and , so that then , hence without loss of generality and hence and thus . Suppose now that 5. holds, and let be left ideals such that . Suppose that there existed so that and . Then still , a contradiction. Note finally that is trivial.