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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?

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I think I can now answer my own question:

The lower-bounds of the Brown-McCoy radical are those ideals that used to be called $G$-rings in the original papers, consisting of elements such that $a\in R(1-a)+(1-a)R$; equivalently, the ideal generated by $1-a$ is $R$.

The upper-bounds of the nilradical are the prime ideals that are not contained in any nil ideal.


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