A taxonomy of 2-primal rings

Various conditions on a noncommutative ring imply that it is 2-primal (i.e., the ringʹs prime radical coincides with the set of nilpotent elements of the ring). We will examine several such conditions and show that their known interdependencies are their only ones. Of particular interest will be the (PS I) condition on a ring (i.e., every factor ring modulo the right annihilator of a principal right ideal is 2-primal). We will see that even within a fairly narrow class of rings, (PS I) is a strictly stronger condition than 2-primal. We will show that the (PS I) condition is left-right asymmetric. We will also study the interplay between various types of semilocal rings and various types of 2-primal rings. The Köthe Conjecture will make a cameo appearance. In Section 6, we will examine subideals of prime ideals of commutative rings that are invariant under derivations.