R. A proper ideal I of the ring R is called a semiprime ideal if it is an intersection of prime ideals of R. A proper ideal P of the ring R is called a left primitive ideal if it is the annihilator of a simple left R-module. P implies a P or b P, for any a,b R. 11.1.4. The annihilator of a prime module is a prime ideal.
Any maximal ideal is primitive, and any primitive ideal is prime.
In a left Artinian ring, the notions of maximal ideal, primitive ideal, and prime ideal coincide. I implies a I, for all a R. 11.1.8. Let R M be a left R-module. Let M be a left R-module. The Jacobson radical 11.2.1.