| simple | → | primitive | → | prime |
| ↓? | ↓ | ↓ | ||
| semisimple | → | semiprimitive | → | semiprime |
Let's take the horizontal arrows first. If R is simple then let M be R mod a maximal left ideal. The annihilator of M is an ideal, and it cannot be R, hence it is 0, and M is faithful. That takes care of the first arrow.
For the next arrow, let M be a faithful simple left R module. For any nonzero ideal H in r, H×M = M. Thus (H*J)×M = H×(J×M) = H×M = M, hence H*J is nonzero. If H*J = 0 then H or J is 0; hence R is a prime ring.
If R is a semisimple ring then it is the finite direct product of simple rings. A maximal left ideal in such a ring is maximal in one component, crossed with all the other component rings. The intersection of all such maximal left ideals becomes a two sided ideal in this component. Since each component is simple, the resulting ideal is 0. This happens in every component, hence R is jacobson semisimple, and semiprimitive.
Let R be semiprimitive and let M be the direct sum of all left simple modules. Let H be a nontrivial ideal, and let x be a nonzero element of H. Since R is jacobson semisimple, x is outside some left ideal, and x does not kill one of the simple submodules Mi of M. In fact x×Mi = Mi, since Mi is simple. The same holds for x*x. Thus x*x is nonzero, and H2 is nonzero, and R is semiprime.
Let's take the vertical arrows, right to left. Prime implies semiprime, right from the definition. Similarly, primitive implies semiprimitive, as described in the introduction. However, the left arrow is problematic. It does not hold in certain situations.
If R is not left artinian, it could be a simple ring that is not semisimple, whence the first vertical arrow fails.
On the other hand, assume R exhibits dcc on its left principal ideals. If you are also given semiprimitive, which is the same as jacobson semisimple, then R becomes semisimple. The first arrow on the bottom row becomes a double arrow.
In fact, the entire bottom row is equivalent, courtesy of a theorem from nil radicals, that asserts semiprime → semisimple.
Simple is also jacobson semisimple, so the left vertical arrow is valid. But we can't go backwards. The direct product of two fields is semisimple, and artinian, but not simple. It's not even a prime ring.
Finally, dcc implies a minimal left ideal, and looking ahead, prime implies left primitive. In fact, it allows us to flip parity, over to right primitive if we wish. Also, primitive implies simple. The top row is an equivalence, and the bottom row is too.
| simple | ⇔ | primitive | ⇔ | prime |
| ↓ | ↓ | ↓ | ||
| semisimple | ⇔ | semiprimitive | ⇔ | semiprime |
Without dcc, simple becomes stronger than primitive. Our ring of endomorphisms is primitive, but not simple (assumeing V is infinite dimensional). R contains an ideal of matrices with finitely many nonzero rows. Thus the first arrow on top remains an implication. You might want to go back and show the other arrows are unidirectional as well.