In the lattice of ideals, what are the lowerbounds of the prime ideals?

Question

Take the lattice of ideals in a non-commutative ring with 1. It is well known that the lowerbounds of all the maximal ideals are the superfluous ideals. Is there a similar characterization for the lowerbounds of all the prime ideals? They must be close to the sum of nilpotent ideals but I can't seem to prove it.

Answer 1

The situation for the maximal ideals is clear cut since the greatest lower bound (their intersection, the Brown-McCoy radical) happens to be the biggest superfluous ideal of the ring. Any ideal below it is also superfluous, and it contains all superfluous ideals.

Following this, we consider the intersection of all prime ideals (The "lower nilradical", "prime radical" or "Baer radical").

It's known (you can check Lambek's Lectures on rings and modules) that the prime radical of a ring consists of all its strongly nilpotent elements. (The definition of $a$ being "strongly nilpotent" is that every sequence beginning with $a_1=a$ such that $a_{i+1}\in a_nRa_n$ has a term which is zero, so that it terminates in zeros.)

If we define a "strongly nil ideal" to be one in which all elements are strongly nilpotent, then the prime radical is the largest strongly nil ideal, and it is obvious that all ideals contained within it are strongly nil ideals, and it's clear that all strongly nil ideals are contained in the prime radical.

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The sum of all nilpotent ideals is obviously contained inside the prime radical, but it may be strictly smaller.