The main radicals of a non-commutative ring (with 1) are the Sum of all nilpotent ideals $\subseteq$ Prime radical $\subseteq$ Nil radical $\subseteq$ Jacobson radical $\subseteq$ Brown-McCoy radical.

Some of these have lowerbound and/or upperbound characterizations. For example, the lb/ub of

- the prime radical are the strong nilpotents and the semi-prime ideals;
- the Jacobson are the quasi-regulars (also known as quasi-nilpotents) and the maximal left ideals;

Is there a similar characterization of the lb of the Brown-McCoy radical as some weak form of nilpotents? And similarly for the ub ideals of the nilradical and the ``SumNilpotent'' radical?