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For questions involving prime ideals in commutative or noncommutative rings.
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What are the upperbounds of the Nil radical?
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 …
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What are the upperbounds of the Nil radical?
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. …